From Metrics to Axioms
Throughout our study of metric spaces,
we have relied on a distance function \(d\) to define open sets, continuity, convergence,
and compactness. Yet a recurring observation has emerged: many of these properties depend
not on the specific numerical values of \(d\), but only on which sets are declared open.
In metric spaces, we noted that the collection of open subsets
determined by \(d\) forms a structure called the topology of \((X, d)\). In
Continuity,
we showed that a function is continuous if and only if preimages of open sets are open — a characterization that
never mentions \(\epsilon\) or \(\delta\) directly. In Homeomorphism,
we identified topological invariants: properties preserved under bijections that
respect open sets in both directions.
The natural question is: can we build a theory of "spaces" starting directly from open sets, without assuming a metric at all?
The answer is yes, and this generalization is not merely aesthetic. There are important spaces in mathematics and computer science — function
spaces with weak topologies, quotient spaces arising from symmetry groups, and configuration spaces in robotics — that carry a natural notion
of "nearness" but no natural metric. To study these spaces, we need a framework that retains the power of open-set reasoning
while discarding the assumption that distances exist.
The Three Axioms
Let us extract the essential properties of open sets in metric spaces. We defined a set \(U\) to be open if every point of \(U\)
is an interior point — equivalently, if \(U\) contains an open ball around each of its points. From this definition, three structural
properties follow.
Theorem: Properties of Open Sets in Metric Spaces
Let \((X, d)\) be a metric space, and let \(\tau_d\) denote the collection of
all open subsets of \(X\). Then:
- \(\emptyset \in \tau_d\) and \(X \in \tau_d\).
- If \(\{U_\alpha\}_{\alpha \in A}\) is any collection of sets in \(\tau_d\),
then \(\displaystyle\bigcup_{\alpha \in A} U_\alpha \in \tau_d\).
- If \(U_1, U_2, \ldots, U_n \in \tau_d\) is a finite collection,
then \(\displaystyle\bigcap_{k=1}^{n} U_k \in \tau_d\).
Proof:
(1) The empty set \(\emptyset\) is open vacuously (there are no points to check).
The whole space \(X\) is open because every point \(x \in X\) satisfies
\(\mathcal{B}(x; r) \subseteq X\) for any \(r > 0\).
(2) Suppose each \(U_\alpha\) is open and \(x \in \bigcup_\alpha U_\alpha\).
Then \(x \in U_\beta\) for some index \(\beta\), so there exists \(r > 0\) with
\(\mathcal{B}(x; r) \subseteq U_\beta \subseteq \bigcup_\alpha U_\alpha\).
Hence the union is open.
(3) The case \(n = 0\) reduces to \(\bigcap_{k=1}^{0} U_k = X \in \tau_d\) by the
standard convention for empty intersections, which is covered by (1).
For \(n \geq 1\), suppose \(U_1, \ldots, U_n\) are open and \(x \in \bigcap_{k=1}^n U_k\).
For each \(k\), there exists \(r_k > 0\) with \(\mathcal{B}(x; r_k) \subseteq U_k\).
Setting \(r = \min\{r_1, \ldots, r_n\} > 0\) (here we use finiteness — a minimum of
finitely many positive numbers is positive), we get
\(\mathcal{B}(x; r) \subseteq \bigcap_{k=1}^n U_k\).
The finiteness restriction in (3) is essential. Consider the family
\(\{(-1/n, 1/n)\}_{n=1}^{\infty}\) in \(\mathbb{R}\). Each interval is open, but
\(\bigcap_{n=1}^{\infty} (-1/n, 1/n) = \{0\}\), a single point, which is not open
in \(\mathbb{R}\). An infinite intersection of open sets can collapse to something
that is no longer open.
These three properties are the only structure we need. We now promote them
from theorems about metric spaces to axioms defining a new kind of space.
Definition: Topological Space
A topological space is a pair \((X, \tau)\) where \(X\) is a set
and \(\tau\) is a collection of subsets of \(X\), called a topology
on \(X\), satisfying:
- \(\emptyset \in \tau\) and \(X \in \tau\).
- \(\tau\) is closed under arbitrary unions: if
\(\{U_\alpha\}_{\alpha \in A} \subseteq \tau\), then
\(\bigcup_{\alpha \in A} U_\alpha \in \tau\).
- \(\tau\) is closed under finite intersections: if
\(U_1, \ldots, U_n \in \tau\), then
\(\bigcap_{k=1}^{n} U_k \in \tau\).
The elements of \(\tau\) are called open sets.
A subset \(F \subseteq X\) is closed if its complement
\(X \setminus F\) is open. We say that \(\tau\) is a topology
on \(X\), and that \((X, \tau)\) is a topological space.
Every metric space \((X, d)\) gives rise to a topological space \((X, \tau_d)\),
but the converse is far from true. There exist topological spaces that cannot be
realized as metric spaces — we will see a concrete example shortly. This is
precisely the power of the abstraction: it captures a strictly larger universe of spaces.
Neighborhoods, Closure, and Interior
In metric spaces, we defined closure, interior, and boundary using the distance function.
In a general topological space, we no longer have distances, but we can reformulate each concept purely
in terms of open sets.
Definition: Neighborhood
Let \((X, \tau)\) be a topological space and \(x \in X\).
A neighborhood of \(x\) is any subset
\(N \subseteq X\) such that there exists an open set \(U \in \tau\)
with \(x \in U \subseteq N\). If \(N\) is itself open, we call it
an open neighborhood.
In many topology texts (following Munkres's convention), a neighborhood
need not be open — it only needs to contain an open set around
the point. Other authors, including Lee's Introduction to Smooth
Manifolds, reserve "neighborhood" for open sets; in that convention,
what we here call a "neighborhood" would be described as "a set containing
a neighborhood of \(x\)." We will use the more permissive convention
throughout. In practice, most arguments pass immediately to an open
neighborhood contained in \(N\), so the distinction rarely affects proofs.
In a metric space, every neighborhood contains an open ball, and every open ball
is a neighborhood, so the two notions coincide. In a general topological space,
neighborhoods take over the role that open balls played in metric spaces.
Definition: Interior, Closure, and Boundary
Let \((X, \tau)\) be a topological space and \(S \subseteq X\).
- The interior of \(S\) is
\[
S^\circ = \operatorname{int}(S) = \bigcup \{ U \in \tau : U \subseteq S \},
\]
the largest open set contained in \(S\).
- The closure of \(S\) is
\[
\overline{S} = \operatorname{cl}(S) = \bigcap \{ F \subseteq X : F \supseteq S \text{ and } F \text{ is closed} \},
\]
the smallest closed set containing \(S\).
- The boundary of \(S\) is
\(\partial S = \overline{S} \setminus S^\circ\).
The "largest" and "smallest" qualifiers require justification. For the interior:
\(S^\circ\) is open because it is a union of open sets (axiom 2); since each
\(U\) in the union satisfies \(U \subseteq S\), the union is also contained in \(S\);
and any open \(U \subseteq S\) appears in the family being unioned, so \(U \subseteq S^\circ\).
These three properties — open, contained in \(S\), and containing every open subset of \(S\) —
determine \(S^\circ\) uniquely (any two sets satisfying them contain each other), justifying the article "the largest."
For the closure, we must first confirm that arbitrary intersections of closed sets are closed.
By De Morgan's laws, if each \(F_\alpha\) is closed, then
\(X \setminus \bigcap_\alpha F_\alpha = \bigcup_\alpha (X \setminus F_\alpha)\)
is a union of open sets, hence open by axiom 2, so \(\bigcap_\alpha F_\alpha\) is closed.
The family in the defining intersection is nonempty (it contains \(X\) itself, which is closed and contains \(S\)),
so \(\overline{S}\) is a well-defined closed set containing \(S\), and any closed \(F \supseteq S\) appears
in the intersection, giving \(\overline{S} \subseteq F\). By the same uniqueness argument as for \(S^\circ\),
these properties single out \(\overline{S}\) as the smallest such closed set.
These definitions are consistent with the metric-space versions from
Metric Spaces.
In a metric space, one can show that the interior \(S^\circ\) consists of all points
having an open ball within \(S\), and the closure \(\overline{S}\) consists of all points
whose every open ball intersects \(S\). The general definitions above recover
these ball-based characterizations as a special case, without requiring a metric
in the definition itself.
The closure and interior admit a neighborhood-based description that will be the
main working tool in later sections. This characterization replaces the role of
"every open ball intersects \(S\)" from metric-space theory.
Theorem: Neighborhood Characterization of Closure and Interior
Let \((X, \tau)\) be a topological space and \(S \subseteq X\). Then:
- \(x \in S^\circ\) if and only if some open neighborhood of \(x\) is contained in \(S\).
- \(x \in \overline{S}\) if and only if every open neighborhood of \(x\) intersects \(S\).
Proof:
(1) (\(\Rightarrow\)) If \(x \in S^\circ = \bigcup\{U \in \tau : U \subseteq S\}\), then \(x \in U\) for some open \(U \subseteq S\); this \(U\) is an open neighborhood of \(x\) contained in \(S\).
(\(\Leftarrow\)) If \(U\) is an open neighborhood of \(x\) with \(U \subseteq S\), then \(U\) appears in the union defining \(S^\circ\), so \(x \in U \subseteq S^\circ\).
(2) We prove the contrapositive: \(x \notin \overline{S}\) if and only if some open neighborhood of \(x\) is disjoint from \(S\).
(\(\Rightarrow\)) Suppose \(x \notin \overline{S}\). Then \(x\) lies in the open set \(U = X \setminus \overline{S}\), and since \(S \subseteq \overline{S}\), we have \(U \cap S = \emptyset\).
(\(\Leftarrow\)) Suppose some open neighborhood \(U\) of \(x\) is disjoint from \(S\). Then \(X \setminus U\) is closed and contains \(S\), so by the defining intersection, \(\overline{S} \subseteq X \setminus U\). Since \(x \in U\), we have \(x \notin \overline{S}\).
Continuity in Topological Spaces
The topological definition of continuity transfers immediately to this setting.
Definition: Continuous Map
Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces. A function
\(f : X \to Y\) is continuous if for every open set
\(V \in \tau_Y\), the preimage \(f^{-1}(V)\) belongs to \(\tau_X\).
This is identical to the global topological characterization from Continuity,
now stated without the assumption that \(X\) and \(Y\) are metric spaces. Similarly, a homeomorphism is a bijection
\(f : X \to Y\) such that both \(f\) and \(f^{-1}\) are continuous — exactly the definition from
Homeomorphism, lifted to this generality.
Fundamental Examples
To build intuition for the flexibility (and potential pathology) of topological spaces, we consider several examples.
The first two are extreme cases that bracket every possible topology on a given set.
Example: Discrete and Indiscrete Topologies
Let \(X\) be any nonempty set.
- The discrete topology on \(X\) is
\(\tau_{\text{disc}} = \mathcal{P}(X)\), the power set. Every subset
is open. This is the topology induced by the discrete metric
\(d(x,y) = \begin{cases} 0 & x=y \\ 1 & x \neq y \end{cases}\),
since every singleton \(\{x\} = \mathcal{B}(x; 1/2)\) is open,
and arbitrary unions of singletons yield all subsets.
- The indiscrete topology (or trivial topology) on \(X\) is
\(\tau_{\text{ind}} = \{\emptyset, X\}\). Only the empty set and the
whole space are open. If \(|X| \geq 2\), this topology is
not metrizable: for any metric \(d\) on \(X\),
pick distinct \(x, y \in X\) and let \(r = d(x,y)/2 > 0\). Then
\(\mathcal{B}(x; r)\) is an open set in the metric topology \(\tau_d\)
containing \(x\) but not \(y\), so \(\mathcal{B}(x; r) \notin \{\emptyset, X\}\).
Hence \(\tau_d \neq \tau_{\text{ind}}\), and no metric induces \(\tau_{\text{ind}}\).
For any topology \(\tau\) on \(X\), we have \(\tau_{\text{ind}} \subseteq \tau \subseteq \tau_{\text{disc}}\).
When \(\tau \subseteq \tau'\), we say \(\tau\) is coarser (fewer open sets) and \(\tau'\) is
finer (more open sets). A finer topology declares more sets to be open, making it easier for functions
into the space to be continuous but harder for functions out of the space to be continuous.
Example: The Cofinite Topology
Let \(X\) be any set. Define
\[
\tau_{\text{cof}} = \{U \subseteq X : X \setminus U \text{ is finite}\} \cup \{\emptyset\}.
\]
This is a topology. Axiom 1 is immediate: \(\emptyset\) is explicitly adjoined, and \(X \setminus X = \emptyset\) is finite, so \(X \in \tau_{\text{cof}}\).
For arbitrary unions: if all \(U_\alpha = \emptyset\), the union is
\(\emptyset \in \tau_{\text{cof}}\). Otherwise, some \(U_\beta \neq \emptyset\), so by definition
\(X \setminus U_\beta\) is finite. Then
\(X \setminus \bigcup_\alpha U_\alpha = \bigcap_\alpha (X \setminus U_\alpha) \subseteq X \setminus U_\beta\), which is finite.
For finite intersections: if any \(U_k = \emptyset\), then \(\bigcap_{k=1}^n U_k = \emptyset \in \tau_{\text{cof}}\).
Otherwise, each \(X \setminus U_k\) is finite, and \(X \setminus \bigcap_{k=1}^n U_k = \bigcup_{k=1}^n (X \setminus U_k)\)
is a finite union of finite sets, which is finite.
On a finite set, the cofinite topology coincides with the discrete topology.
On an infinite set, it is strictly coarser than the discrete topology and
is not metrizable: if \(U\) and \(V\) are nonempty
open sets, then \(X \setminus (U \cap V) = (X \setminus U) \cup (X \setminus V)\)
is a finite union of finite sets, hence finite, so \(U \cap V\) is
nonempty. Since no two nonempty open sets are disjoint, the space
is not Hausdorff; since every metric space is Hausdorff
(proved as
Every Metric Space is Hausdorff below),
\(\tau_{\text{cof}}\) cannot arise from any metric.
Why Non-Metrizable Spaces Matter
At this point one might ask: if metric spaces already cover \(\mathbb{R}^n\),
function spaces, and most spaces in applied mathematics, why bother with
non-metrizable spaces at all?
The answer lies in our own recent work.
In Weak Topologies & Banach-Alaoglu, we
proved that the closed unit ball of the dual space \(\mathcal{X}^*\) is compact in the weak* topology.
The proof relied on embedding \(\mathcal{X}^*\) into a product space \(\prod_{x \in \mathcal{X}} [-\|x\|, \|x\|]\)
and invoking Tychonoff's theorem. But we used the product topology and Tychonoff's theorem as
unproven black boxes — tools from general topology that metric-space theory alone cannot provide.
This chapter repays that debt: we will define the product topology rigorously in
Product Topology and state Tychonoff's theorem
with its proper context.
Bases & Countability
Specifying a topology by listing all open sets is usually impractical —
the standard topology on \(\mathbb{R}\) contains uncountably many open sets.
Instead, we describe a topology through a smaller collection of "building blocks"
from which all open sets can be constructed.
Basis for a Topology
Definition: Basis
Let \((X, \tau)\) be a topological space. A collection
\(\mathcal{B} \subseteq \tau\) is a basis for \(\tau\) if
every open set \(U \in \tau\) can be written as a union of elements
of \(\mathcal{B}\). Equivalently, for every \(U \in \tau\) and every
\(x \in U\), there exists \(B \in \mathcal{B}\) with \(x \in B \subseteq U\).
We already know the prototypical example: in a metric space \((X, d)\),
the collection of all open balls
\(\mathcal{B} = \{\mathcal{B}(x; r) : x \in X,\; r > 0\}\)
is a basis for the metric topology \(\tau_d\). This was stated in
Metric Spaces, where
we noted that "every open set can be expressed as a union of open balls."
The power of the basis concept is that we can also go in the reverse direction:
start with a collection of sets satisfying certain properties, and
generate a topology from them.
Theorem: Basis Criterion
Let \(X\) be a set and \(\mathcal{B}\) a collection of subsets of \(X\)
satisfying:
- (Covering) For every \(x \in X\), there exists
\(B \in \mathcal{B}\) with \(x \in B\).
- (Refinement) If \(B_1, B_2 \in \mathcal{B}\) and
\(x \in B_1 \cap B_2\), then there exists \(B_3 \in \mathcal{B}\)
with \(x \in B_3 \subseteq B_1 \cap B_2\).
Then the collection
\[
\tau_{\mathcal{B}} = \left\{ U \subseteq X : \text{for every } x \in U,
\text{ there exists } B \in \mathcal{B} \text{ with } x \in B \subseteq U \right\}
\]
is a topology on \(X\), and \(\mathcal{B}\) is a basis for \(\tau_{\mathcal{B}}\).
Proof:
We verify the three axioms. The empty set \(\emptyset\) belongs to
\(\tau_{\mathcal{B}}\) vacuously. The whole space \(X\) belongs because
the covering condition guarantees every point lies in some basis element.
For arbitrary unions: if each \(U_\alpha \in \tau_{\mathcal{B}}\) and
\(x \in \bigcup_\alpha U_\alpha\), then \(x \in U_\beta\) for some
\(\beta\), so there exists \(B \in \mathcal{B}\) with
\(x \in B \subseteq U_\beta \subseteq \bigcup_\alpha U_\alpha\).
For finite intersections, it suffices to handle the case of two sets
(the general case follows by induction). Suppose
\(U_1, U_2 \in \tau_{\mathcal{B}}\) and \(x \in U_1 \cap U_2\).
Then there exist \(B_1, B_2 \in \mathcal{B}\) with
\(x \in B_1 \subseteq U_1\) and \(x \in B_2 \subseteq U_2\).
By the refinement condition, there exists \(B_3 \in \mathcal{B}\) with
\(x \in B_3 \subseteq B_1 \cap B_2 \subseteq U_1 \cap U_2\).
Finally, \(\mathcal{B}\) is a basis for \(\tau_{\mathcal{B}}\). Each \(B \in \mathcal{B}\) lies in \(\tau_{\mathcal{B}}\): for any \(x \in B\), \(B\) itself witnesses \(x \in B \subseteq B\). Conversely, let \(U \in \tau_{\mathcal{B}}\); for each \(x \in U\), pick \(B_x \in \mathcal{B}\) with \(x \in B_x \subseteq U\). Then \(U = \bigcup_{x \in U} B_x\): the inclusion \(\supseteq\) holds since each \(B_x \subseteq U\), and \(\subseteq\) holds since every \(x \in U\) lies in \(B_x\). Hence every open set is a union of basis elements.
The basis criterion gives us a systematic recipe: to construct a topology,
it suffices to specify a basis and verify two conditions, rather than
checking the full topology axioms on an unwieldy collection. Moreover,
\(\tau_{\mathcal{B}}\) is the unique topology on \(X\) admitting \(\mathcal{B}\) as a basis:
any topology with \(\mathcal{B}\) as basis must contain exactly the unions of elements of \(\mathcal{B}\),
and this family coincides with \(\tau_{\mathcal{B}}\) by the argument above.
Countability Axioms
Not all topological spaces are equally well-behaved. The countability
axioms impose size constraints on the topology that ensure many familiar
tools from metric-space analysis remain valid.
Definition: First Countable and Second Countable
Let \((X, \tau)\) be a topological space.
- \(X\) is first countable if every point \(x \in X\)
has a countable neighborhood basis: a countable
collection \(\{U_n\}_{n=1}^{\infty}\) of open neighborhoods of \(x\)
such that every neighborhood of \(x\) contains some \(U_n\).
- \(X\) is second countable if \(\tau\) itself
admits a countable basis: a countable collection
\(\mathcal{B}\) that is a basis for \(\tau\).
When \(\{U_n\}\) is a countable neighborhood basis at \(x\), replacing
\(U_n\) by \(V_n := U_1 \cap \cdots \cap U_n\) (a finite intersection
of open sets, hence open, and still a neighborhood of \(x\)) produces a
nested countable neighborhood basis \(V_1 \supseteq V_2 \supseteq \cdots\);
this nested form is convenient for diagonal and sequential arguments.
Second countability is strictly stronger than first countability.
If \(\mathcal{B} = \{B_n\}\) is a countable basis for \(\tau\) and \(x \in X\),
then \(\{B_n : x \in B_n\}\) is a countable collection of open neighborhoods of \(x\);
for any neighborhood \(U\) of \(x\), pick an open \(V\) with \(x \in V \subseteq U\),
then pick \(B_n \in \mathcal{B}\) with \(x \in B_n \subseteq V\) by the basis property.
Thus every second countable space is first countable.
The converse fails: consider an uncountable set \(X\) with the discrete topology.
Each singleton \(\{x\}\) is open, and \(\{\{x\}\}\) is itself a countable (indeed finite)
neighborhood basis at \(x\), so \(X\) is first countable. But \(X\) is not second countable:
if \(\mathcal{B}\) were any basis, then for each \(x\) the open set \(\{x\}\) must be a
union of basis elements, yet the only \(B \in \mathcal{B}\) with \(x \in B \subseteq \{x\}\)
is \(B = \{x\}\) itself. Hence every singleton must belong to \(\mathcal{B}\), forcing
\(|\mathcal{B}| \geq |X|\), which is uncountable.
Example: Countability in Familiar Spaces
Every metric space is first countable: at any point \(x\), the collection
\(\{\mathcal{B}(x; 1/n)\}_{n=1}^{\infty}\) is a countable neighborhood basis.
The space \(\mathbb{R}^n\) with the standard metric is second countable:
the collection of open balls with rational centers and rational radii,
\[
\mathcal{B} = \{\mathcal{B}(\mathbf{q}; r) : \mathbf{q} \in \mathbb{Q}^n,\; r \in \mathbb{Q}^+\},
\]
is countable (a countable product of countable sets is countable) and forms
a basis for the standard topology. Indeed, for any open set \(U\) and any
\(x \in U\), pick \(\varepsilon > 0\) with \(\mathcal{B}(x; \varepsilon) \subseteq U\).
By density of \(\mathbb{Q}^n\) in \(\mathbb{R}^n\) and \(\mathbb{Q}^+\) in
\(\mathbb{R}^+\), choose \(\mathbf{q} \in \mathbb{Q}^n\) with
\(\|x - \mathbf{q}\| < \varepsilon/3\) and \(r \in \mathbb{Q}^+\) with
\(\varepsilon/3 < r < 2\varepsilon/3\). Then \(x \in \mathcal{B}(\mathbf{q}; r)\)
(since \(\|x - \mathbf{q}\| < \varepsilon/3 < r\)), and for any
\(y \in \mathcal{B}(\mathbf{q}; r)\) the triangle inequality gives
\(\|y - x\| \leq \|y - \mathbf{q}\| + \|\mathbf{q} - x\| < r + \varepsilon/3 < \varepsilon\),
so \(\mathcal{B}(\mathbf{q}; r) \subseteq \mathcal{B}(x; \varepsilon) \subseteq U\).
Separability and the Lindelöf Property
Second countability has powerful consequences that connect to concepts
we have already encountered.
Definition: Separable Space
A topological space \(X\) is separable if it contains a
countable dense subset: a countable set \(D \subseteq X\) such that
\(\overline{D} = X\).
Theorem: Second Countable Implies Separable
Every second countable space is separable.
Proof:
Let \(\mathcal{B} = \{B_n\}_{n=1}^{\infty}\) be a countable basis.
For each index \(n\) with \(B_n \neq \emptyset\), pick a point \(x_n \in B_n\),
and set \(D = \{x_n : B_n \neq \emptyset\}\); this is countable as a subset of a
countable family. To see \(\overline{D} = X\), fix \(x \in X\) and any open neighborhood
\(U \ni x\); the basis definition gives \(B_n \in \mathcal{B}\) with \(x \in B_n \subseteq U\),
so in particular \(B_n \neq \emptyset\) and \(x_n \in B_n \subseteq U\), witnessing
\(U \cap D \neq \emptyset\). By the
Neighborhood Characterization of Closure and Interior,
\(x \in \overline{D}\); hence \(\overline{D} = X\).
Theorem: Lindelöf Property
Every second countable space is Lindelöf: every open
cover has a countable subcover.
Proof:
Let \(\{U_\alpha\}_{\alpha \in A}\) be an open cover of \(X\), and let
\(\mathcal{B} = \{B_n\}_{n=1}^{\infty}\) be a countable basis.
Step 1 (selecting a countable subfamily of the basis).
Define
\[
\mathcal{B}' = \{B_n \in \mathcal{B} : B_n \subseteq U_\alpha \text{ for some } \alpha \in A\}.
\]
This is a subfamily of the countable family \(\mathcal{B}\), hence countable. Note that
the definition depends only on the cover \(\{U_\alpha\}\), not on any choice of indices.
Step 2 (\(\mathcal{B}'\) covers \(X\)).
Fix \(x \in X\). Since \(\{U_\alpha\}\) covers \(X\), pick some \(\alpha_0\) with
\(x \in U_{\alpha_0}\). By the basis property applied to the open set \(U_{\alpha_0}\),
there exists \(B_n \in \mathcal{B}\) with \(x \in B_n \subseteq U_{\alpha_0}\).
This \(B_n\) satisfies the defining condition of \(\mathcal{B}'\), so \(B_n \in \mathcal{B}'\),
and \(x \in B_n\). Hence \(\mathcal{B}'\) covers \(X\).
Step 3 (extracting the countable subcover).
For each \(B_n \in \mathcal{B}'\), pick one index \(\alpha(n) \in A\) with
\(B_n \subseteq U_{\alpha(n)}\); such an index exists by the defining condition of \(\mathcal{B}'\).
(This selection is an application of the axiom of countable choice — a mild
choice principle standard in analysis, which we assume throughout.)
The family \(\{U_{\alpha(n)} : B_n \in \mathcal{B}'\}\) is countable.
To see it covers \(X\): given \(x \in X\), Step 2 provides \(B_n \in \mathcal{B}'\) with \(x \in B_n\),
and then \(x \in B_n \subseteq U_{\alpha(n)}\). Thus \(\{U_{\alpha(n)}\}\) is a countable subcover of
\(\{U_\alpha\}\).
Compactness in General Topological Spaces
The definition of compactness from
Compactness
transfers verbatim to the topological setting, using open covers in the topology
rather than open covers defined via a metric. A topological space \((X, \tau)\) is
compact if every open cover of \(X\) has a finite subcover.
A subset \(K \subseteq X\) is compact if it is compact as a topological space
under the subspace topology (defined formally in
Subspace Topology below); equivalently,
every collection of open sets in \(X\) whose union contains \(K\) admits a finite
subcollection still covering \(K\). The latter ambient form is the one
used in proofs and reduces to the metric-space definition when \(\tau\) comes from
a metric. We use this extended notion freely below.
The Lindelöf property bridges second countability and
compactness:
a second countable space is compact if and only if every countable
open cover has a finite subcover. The "only if" direction is immediate
from the definition of compactness (any countable open cover is, in particular,
an open cover). For "if", given any open cover, extract a countable subcover
by the Lindelöf property, then apply the hypothesis to obtain a finite subcover.
A Remark on Sequences and Nets
In our study of Limits & Convergence,
sequences were the primary tool: a sequence \((x_n)\) converges to \(x\) in a metric space
if \(d(x_n, x) \to 0\). In a general topological space, we say \(x_n \to x\) if every neighborhood
of \(x\) contains all but finitely many terms of the sequence.
However, in general topological spaces, sequences alone are not sufficient
to detect all topological structure. For instance, there exist non-closed sets
\(S\) in certain topological spaces such that no sequence in \(S\) converges
to a point outside \(S\) — the sequential closure is strictly smaller than the
topological closure. A concrete example is \(X = \mathbb{R}\) with the
cocountable topology (a set is open iff its complement is countable or
empty): taking \(S = \mathbb{R} \setminus \{0\}\), any convergent sequence in \(S\)
is eventually constant, so no sequence in \(S\) converges to \(0\); yet every
open neighborhood of \(0\) has countable complement and so must meet \(S\), giving
\(0 \in \overline{S}\). The correct generalization is a net:
a function from a directed set (a generalization of \(\mathbb{N}\)
with its usual ordering) into the space. In the theory of nets,
a set is closed if and only if it contains all limits of convergent nets,
recovering the full power of the closure operation.
The reassuring fact is: in first countable spaces, sequences suffice.
A set is closed if and only if it is sequentially closed, and a function is
continuous if and only if it preserves sequential convergence.
Since every metric space is first countable (and every second countable space
is first countable), all sequence-based arguments from
Limits & Convergence
through Compactness
remain valid in the spaces we will encounter on the path to manifolds.
We will not develop the full theory of nets here, but awareness of this subtlety
is important when working with spaces that are not first countable, such as
certain function spaces with the weak* topology.
Hausdorff & Separation
The axioms of a topological space are remarkably permissive —
they allow spaces where distinct points are "topologically
indistinguishable." To exclude such pathologies, we impose
separation axioms: conditions that ensure the topology
has enough open sets to tell points apart. The most important of these
for our purposes is the Hausdorff condition.
The Separation Hierarchy
Definition: \(T_1\) and \(T_2\) (Hausdorff) Spaces
Let \((X, \tau)\) be a topological space.
- \(X\) is a \(T_1\)-space if for every pair of
distinct points \(x \neq y\), there exists an open set \(U\)
containing \(x\) but not \(y\). Equivalently, every singleton
\(\{x\}\) is a closed set.
- \(X\) is a \(T_2\)-space (or Hausdorff)
if for every pair of distinct points \(x \neq y\), there exist
disjoint open sets \(U\) and \(V\) with \(x \in U\) and \(y \in V\).
The equivalence stated for \(T_1\) deserves brief justification. Suppose the
separating-open-set form holds. Fix \(x \in X\); for each \(y \neq x\), the
definition (applied with roles swapped) gives an open \(U_y\) containing \(y\)
but not \(x\). Then \(X \setminus \{x\} = \bigcup_{y \neq x} U_y\) is a union of
open sets, hence open, so \(\{x\}\) is closed. Conversely, if every singleton is closed
and \(x \neq y\), then \(U = X \setminus \{y\}\) is open, contains \(x\), and excludes \(y\).
Every \(T_2\) space is \(T_1\) (if \(U\) and \(V\) are disjoint with
\(x \in U\) and \(y \in V\), then \(U\) contains \(x\) but not \(y\)).
The converse fails: the cofinite topology on an infinite set is \(T_1\)
(every finite set is closed, so every singleton is closed)
but not Hausdorff (any two nonempty open sets have nonempty intersection,
since their complements are finite).
Theorem: Every Metric Space is Hausdorff
If \((X, d)\) is a metric space, then \((X, \tau_d)\) is Hausdorff.
Proof:
Let \(x \neq y\) in \(X\), and set \(r = d(x, y)/2 > 0\).
Then \(U = \mathcal{B}(x; r)\) and \(V = \mathcal{B}(y; r)\) are disjoint
open sets: if \(z \in U \cap V\), the triangle inequality gives
\(d(x, y) \leq d(x, z) + d(z, y) < r + r = d(x, y)\),
a contradiction.
Why Hausdorff Matters
The Hausdorff condition is not a mere technicality — it has
profound consequences for the behavior of limits and compact sets.
Theorem: Uniqueness of Limits
In a Hausdorff space, limits of convergent sequences are unique:
if \(x_n \to x\) and \(x_n \to y\), then \(x = y\).
Proof:
Suppose \(x \neq y\). By the Hausdorff condition, choose disjoint open sets
\(U \ni x\) and \(V \ni y\). Using the
topological definition of convergence —
every neighborhood of the limit contains all but finitely many terms —
there exists \(N_1\) with \(x_n \in U\) for all \(n \geq N_1\) (since \(x_n \to x\)),
and \(N_2\) with \(x_n \in V\) for all \(n \geq N_2\) (since \(x_n \to y\)).
For \(n \geq \max(N_1, N_2)\),
we get \(x_n \in U \cap V = \emptyset\), a contradiction.
In a non-Hausdorff space, a sequence can converge to multiple limits
simultaneously — a fundamental obstruction for any theory in which limits
must be well-defined (classical calculus, optimization, convergence of algorithms).
Theorem: Compact Subsets of Hausdorff Spaces are Closed
If \(X\) is a Hausdorff space and \(K \subseteq X\) is compact,
then \(K\) is closed.
Proof:
If \(K = \emptyset\), it is closed trivially; assume \(K \neq \emptyset\).
We show that \(X \setminus K\) is open. Let \(x \in X \setminus K\).
For each \(y \in K\), the Hausdorff condition provides disjoint open sets
\(U_y \ni x\) and \(V_y \ni y\). The collection \(\{V_y\}_{y \in K}\)
covers \(K\), so by compactness there exist \(y_1, \ldots, y_n\) with
\(K \subseteq V_{y_1} \cup \cdots \cup V_{y_n}\).
Set \(U = U_{y_1} \cap \cdots \cap U_{y_n}\).
Then \(U\) is a finite intersection of open sets containing \(x\),
so \(U\) is an open neighborhood of \(x\). Moreover, \(U \subseteq U_{y_i}\)
for each \(i\), so \(U \cap V_{y_i} \subseteq U_{y_i} \cap V_{y_i} = \emptyset\);
hence \(U\) is disjoint from \(V_{y_1} \cup \cdots \cup V_{y_n} \supseteq K\),
giving \(U \subseteq X \setminus K\). Since \(x\) was arbitrary,
\(X \setminus K\) is open.
This generalizes a fact we relied on in
Compactness:
in \(\mathbb{R}^n\), compact sets are closed and bounded (Heine-Borel).
The "closed" part holds in any Hausdorff space; the "bounded" part requires
additional structure (a metric, or more generally a uniformity) beyond mere topology.
In a general Hausdorff space, compactness
captures the essential meaning of "finite-like" behavior:
compact subsets are closed, and in particular, compact Hausdorff spaces
are as well-behaved as one could hope for.
The Line with Two Origins: A Cautionary Tale
To appreciate why the Hausdorff condition is explicitly required in the
definition of a manifold, consider the following pathological example.
Example: The Line with Two Origins
Take two copies of the real line, \(\mathbb{R} \times \{a\}\) and
\(\mathbb{R} \times \{b\}\), and glue them together by identifying
\((x, a)\) with \((x, b)\) for every \(x \neq 0\).
The formal construction — the quotient topology — is deferred to
Quotient Topology below;
for now we describe its behavior heuristically.
The resulting space \(L\) has a single copy of each nonzero real number, but
two distinct origins: the points \(0_a = (0, a)\) and \(0_b = (0, b)\) are not identified.
Heuristically, each point of \(L\) has a neighborhood that behaves like an
open interval in \(\mathbb{R}\) — the formal statement, which uses the
"locally Euclidean" property of manifolds, belongs to the later manifold
chapters. What matters here is the global failure: \(L\) is
not Hausdorff.
Any open set containing \(0_a\) inherits an interval's worth of
\((x, a)\)-points around zero from the \(a\)-copy, and similarly any
open set containing \(0_b\) inherits \((x, b)\)-points around zero
from the \(b\)-copy. After gluing, these intervals overlap at all
nonzero points in some common interval around zero, so the two
origins cannot be separated by disjoint open sets. A rigorous
verification using the quotient topology appears in
The Hausdorff Warning below.
Since every metric space is Hausdorff (by the theorem above), it follows that
\(L\) is not metrizable.
The line with two origins is locally indistinguishable from \(\mathbb{R}\),
yet globally pathological: sequences can converge to two different limits,
and compact subsets need not be closed. This is exactly the kind of space
that the Hausdorff axiom in the manifold definition is designed to exclude.
When we construct new spaces via quotients in
Quotient Topology below, we will need to verify the Hausdorff
condition carefully — it is not automatically inherited.
Connection to Lie Groups and Geometric Deep Learning
In the theory of Lie groups, one frequently forms quotient spaces
\(G/H\) where \(H\) is a closed subgroup of a Lie group \(G\).
For closed subgroups \(H\) of a topological group \(G\), the quotient
\(G/H\) inherits the Hausdorff property — the formal treatment of this
fact belongs to later Lie group and manifold chapters. The qualitative
lesson is already visible here: closedness of \(H\) is what saves
the quotient from pathologies like the line with two origins
(locally smooth, globally ill-defined).
In Geometric Deep Learning, symmetry groups act on feature spaces,
and the orbit space (the quotient by the group action) must be Hausdorff for the
resulting representations to be well-defined.
The separation axioms are not abstract formalism — they are the
gatekeepers ensuring that "quotienting by symmetry" produces a
meaningful space.
Subspace, Product & Quotient Topologies
With the language of topological spaces, separation, and countability
in hand, we turn to the three fundamental constructions for
building new topological spaces from existing ones. Every manifold
we will encounter is assembled from these operations: submanifolds
inherit the subspace topology, product manifolds like
\(M \times N\) carry the product topology, and orbit spaces like
\(G/H\) are formed via quotients. Mastering these constructions now
is essential for the chapters ahead.
Subspace Topology
Throughout Metric Spaces
through Homeomorphism,
we freely worked with subsets of metric spaces — open intervals in
\(\mathbb{R}\), spheres in \(\mathbb{R}^n\), closed subspaces of
Banach spaces — always assuming that the "natural" topology on a subset
is the one inherited from the ambient space. Let us now make this precise.
Definition: Subspace Topology
Let \((X, \tau)\) be a topological space and \(S \subseteq X\).
The subspace topology (or induced topology)
on \(S\) is
\[
\tau_S = \{ U \cap S : U \in \tau \}.
\]
The pair \((S, \tau_S)\) is called a subspace of \(X\).
Verification:
We check the three axioms. (1) \(\emptyset = \emptyset \cap S \in \tau_S\)
and \(S = X \cap S \in \tau_S\).
(2) If \(\{U_\alpha \cap S\}\) is a family in \(\tau_S\), then
\(\bigcup_\alpha (U_\alpha \cap S) = (\bigcup_\alpha U_\alpha) \cap S \in \tau_S\)
since \(\bigcup_\alpha U_\alpha \in \tau\).
(3) Similarly, a finite intersection
\(\bigcap_{k=1}^n (U_k \cap S) = (\bigcap_{k=1}^n U_k) \cap S \in \tau_S\).
The subspace topology is the coarsest topology making the inclusion
\(\iota : S \hookrightarrow X\) (defined by \(\iota(s) = s\)) continuous,
and it is characterized by a universal property that lets us reduce
continuity of maps into \(S\) to continuity into the ambient space —
a technique we have used implicitly throughout the metric-space chapters.
Theorem: Universal Property of the Subspace Topology
Let \((X, \tau)\) be a topological space and \(S \subseteq X\) with the
subspace topology \(\tau_S\), and let \(\iota : S \hookrightarrow X\)
be the inclusion map.
- \(\iota\) is continuous, and \(\tau_S\) is the coarsest topology on \(S\)
for which \(\iota\) is continuous.
- For any topological space \(Y\) and any function \(f : Y \to S\),
\(f\) is continuous if and only if \(\iota \circ f : Y \to X\) is continuous.
Proof:
(1) For any \(U \in \tau\), we have \(\iota^{-1}(U) = U \cap S \in \tau_S\)
by the definition of \(\tau_S\), so \(\iota\) is continuous. For any other
topology \(\tau'\) on \(S\) making \(\iota\) continuous, the preimage
\(\iota^{-1}(U) = U \cap S\) must lie in \(\tau'\) for every \(U \in \tau\);
since every element of \(\tau_S\) has this form, \(\tau_S \subseteq \tau'\).
(2) (\(\Rightarrow\)) If \(f\) is continuous, then \(\iota \circ f\) is a
composition of continuous maps, hence continuous.
(\(\Leftarrow\)) Suppose \(\iota \circ f\) is continuous. For any
\(V \in \tau_S\), write \(V = U \cap S\) with \(U \in \tau\).
Since \(f\) maps into \(S\), the condition \(f(y) \in V\) reduces to
\(f(y) \in U\); hence
\(f^{-1}(V) = f^{-1}(U \cap S) = (\iota \circ f)^{-1}(U)\), which is open in \(Y\)
by continuity of \(\iota \circ f\). Thus \(f\) is continuous.
Theorem: Inheritance of Properties
Let \(S\) be a subspace of \(X\).
- If \(X\) is Hausdorff, then \(S\) is Hausdorff.
- If \(X\) is first (resp. second) countable, then so is \(S\).
Proof:
Hausdorff. If \(s_1 \neq s_2\) in \(S\), choose
disjoint open sets \(U_1 \ni s_1\) and \(U_2 \ni s_2\) in \(X\).
Then \(U_1 \cap S\) and \(U_2 \cap S\) are disjoint open sets in
\(\tau_S\) separating \(s_1\) and \(s_2\).
First countable. Fix \(s \in S\). By first countability of \(X\),
there is a countable neighborhood basis \(\{U_n\}\) of \(s\) in \(X\).
We claim \(\{U_n \cap S\}\) is a countable neighborhood basis of \(s\) in \(\tau_S\).
Each \(U_n \cap S\) is open in \(\tau_S\) and contains \(s\). Let \(N\) be any
neighborhood of \(s\) in \(S\); by definition there is an open \(V \in \tau_S\)
with \(s \in V \subseteq N\), and we write \(V = W \cap S\) for some \(W \in \tau\)
(so \(s \in W\) since \(s \in V\)). First countability of \(X\) gives an index
\(N'\) with \(U_{N'} \subseteq W\), whence \(U_{N'} \cap S \subseteq W \cap S = V \subseteq N\).
Hence every neighborhood of \(s\) in \(S\) contains some \(U_n \cap S\).
Second countable. If \(\mathcal{B}\) is a countable basis for \(\tau\),
then \(\{B \cap S : B \in \mathcal{B}\}\) is a countable basis for \(\tau_S\):
every \(V \in \tau_S\) equals \(W \cap S\) for some \(W \in \tau\), and
writing \(W = \bigcup_\alpha B_\alpha\) with \(B_\alpha \in \mathcal{B}\) gives
\(V = \bigcup_\alpha (B_\alpha \cap S)\).
Open Maps and Closed Maps
Before defining the product and quotient topologies, we need
a concept that complements continuity. A continuous map pulls
open sets back through preimages; an open map
pushes open sets forward through images.
Definition: Open Map and Closed Map
Let \(f : X \to Y\) be a map between topological spaces.
- \(f\) is an open map if for every open set
\(U \subseteq X\), the image \(f(U)\) is open in \(Y\).
- \(f\) is a closed map if for every closed set
\(F \subseteq X\), the image \(f(F)\) is closed in \(Y\).
Theorem: Homeomorphism Characterization via Open/Closed Maps
Let \(f : X \to Y\) be a continuous bijection between topological spaces.
The following are equivalent:
- \(f\) is a homeomorphism.
- \(f\) is an open map.
- \(f\) is a closed map.
Proof:
\((1) \Leftrightarrow (2)\): Since \(f\) is a bijection,
\(f(U) = (f^{-1})^{-1}(U)\) for every \(U \subseteq X\). Thus
"\(f(U)\) is open for every open \(U\)" is exactly "\(f^{-1}\) is continuous,"
which combined with the continuity of \(f\) means \(f\) is a homeomorphism.
\((2) \Leftrightarrow (3)\): For any bijection \(f\) and any
\(S \subseteq X\), the identity \(f(X \setminus S) = Y \setminus f(S)\) holds
(surjectivity gives \(f(X) = Y\), and injectivity gives
\(f(X \setminus S) = f(X) \setminus f(S)\)). Applying this with \(S\) ranging
over closed subsets \(F \subseteq X\): \(f(F)\) is closed in \(Y\) iff
\(Y \setminus f(F) = f(X \setminus F)\) is open in \(Y\), and \(X \setminus F\)
ranges over all open subsets of \(X\) as \(F\) ranges over all closed subsets.
Thus \(f\) sends closed sets to closed sets if and only if it sends open sets
to open sets.
In general, continuity alone does not imply that a map is open or closed —
these are independent properties. For instance, a constant map \(f : X \to Y\)
with \(f(x) = y_0\) is continuous, and (provided \(\{y_0\}\) is closed in \(Y\) —
e.g., when \(Y\) is \(T_1\)) is also a closed map: the image of any subset of \(X\)
is either \(\emptyset\) or \(\{y_0\}\), both closed. But \(\{y_0\}\) is typically
not open, so constant maps are not open in general. We will see immediately
that projection maps from product spaces are both continuous and open,
while quotient maps are continuous but not necessarily open.
Product Topology
Given topological spaces \((X_1, \tau_1)\) and \((X_2, \tau_2)\),
we want to topologize the Cartesian product \(X_1 \times X_2\) in a way
that respects both factor topologies. The correct construction uses
the basis criterion.
Definition: Product Topology (Finite Case)
Let \((X_1, \tau_1)\) and \((X_2, \tau_2)\) be topological spaces.
The product topology on \(X_1 \times X_2\) is the
topology generated by the basis
\[
\mathcal{B} = \{ U_1 \times U_2 : U_1 \in \tau_1,\; U_2 \in \tau_2 \}.
\]
More generally, for a finite product \(X_1 \times \cdots \times X_n\),
the basis consists of all products
\(U_1 \times \cdots \times U_n\) with each \(U_i \in \tau_i\).
Verification of the Basis Criterion:
(Covering) For any \((x_1, x_2) \in X_1 \times X_2\), the set
\(X_1 \times X_2\) itself is a basis element.
(Refinement) If \((x_1, x_2) \in (U_1 \times U_2) \cap (V_1 \times V_2)\),
then \((x_1, x_2) \in (U_1 \cap V_1) \times (U_2 \cap V_2)\),
which is a basis element contained in the intersection.
The projection maps \(\pi_i : X_1 \times X_2 \to X_i\)
defined by \(\pi_i(x_1, x_2) = x_i\) are the natural "coordinate extraction"
functions. They have two important properties.
Theorem: Projections are Continuous and Open
Each projection \(\pi_i : X_1 \times X_2 \to X_i\) is continuous
and an open map.
Proof:
Continuity: for \(U_1 \in \tau_1\), we have
\(\pi_1^{-1}(U_1) = U_1 \times X_2\), which is open in the product
topology.
Openness: any open \(W \subseteq X_1 \times X_2\) is a union of basis elements
\(W = \bigcup_\alpha (U_{1,\alpha} \times U_{2,\alpha})\). Images commute with unions, so
\[
\pi_1(W) = \bigcup_\alpha \pi_1(U_{1,\alpha} \times U_{2,\alpha}),
\]
where each term equals \(U_{1,\alpha}\) if \(U_{2,\alpha} \neq \emptyset\) and \(\emptyset\)
otherwise; in both cases this is open in \(X_1\). Hence \(\pi_1(W)\) is a union of
open sets in \(X_1\), open by axiom 2. Symmetrically for \(\pi_2\).
The product topology is also characterized by a universal property,
which gives both an abstract description (coarsest topology making projections continuous)
and a working criterion for continuity of maps into the product
(reducing to continuity componentwise).
Theorem: Universal Property of the Product Topology
Let \((X_1, \tau_1)\) and \((X_2, \tau_2)\) be topological spaces, and let
\(\tau_{X_1 \times X_2}\) denote the product topology.
- Coarsest topology. \(\tau_{X_1 \times X_2}\) is the coarsest topology
on \(X_1 \times X_2\) making both projections \(\pi_1, \pi_2\) continuous.
- Component test. For any topological space \(Y\), a function
\(f : Y \to X_1 \times X_2\) written componentwise as
\(f(y) = (f_1(y), f_2(y))\) is continuous if and only if each component
\(f_i = \pi_i \circ f : Y \to X_i\) is continuous.
Proof:
(2) (\(\Rightarrow\)) If \(f\) is continuous, then \(f_i = \pi_i \circ f\)
is a composition of continuous maps, hence continuous.
(\(\Leftarrow\)) Suppose each \(f_i\) is continuous. For any basis element
\(U_1 \times U_2\) with \(U_i \in \tau_i\),
\[
f^{-1}(U_1 \times U_2) = \{y \in Y : f_1(y) \in U_1,\; f_2(y) \in U_2\}
= f_1^{-1}(U_1) \cap f_2^{-1}(U_2),
\]
which is open in \(Y\) as an intersection of two open sets. Since preimages
commute with unions and every open set in \(X_1 \times X_2\) is a union of
basis elements, \(f\) is continuous.
(1) Any topology \(\tau'\) on \(X_1 \times X_2\) making both projections continuous
must contain \(\pi_i^{-1}(U_i) = U_i \times X_j\) for every \(U_i \in \tau_i\)
(where \(j \neq i\)), hence all finite intersections
\(\pi_1^{-1}(U_1) \cap \pi_2^{-1}(U_2) = U_1 \times U_2\), which are exactly the
basis of \(\tau_{X_1 \times X_2}\). Since \(\tau'\) is closed under arbitrary unions,
it contains every union of these basis elements — that is, every set in
\(\tau_{X_1 \times X_2}\). Hence \(\tau_{X_1 \times X_2} \subseteq \tau'\). Finally,
\(\tau_{X_1 \times X_2}\) itself makes the projections continuous: the computation
\(\pi_i^{-1}(U_i) = U_i \times X_j\) produces a basis element, which lies in the
product topology.
Theorem: Product Preserves Hausdorff and Second Countability
Let \(X_1, \ldots, X_n\) be topological spaces.
- If each \(X_i\) is Hausdorff, then
\(X_1 \times \cdots \times X_n\) is Hausdorff.
- If each \(X_i\) is second countable, then
\(X_1 \times \cdots \times X_n\) is second countable.
Proof (for two factors):
Hausdorff: suppose \((x_1, x_2) \neq (y_1, y_2)\). Then \(x_i \neq y_i\)
for some \(i\). Choose disjoint open sets \(U \ni x_i\) and
\(V \ni y_i\) in \(X_i\). Then \(\pi_i^{-1}(U)\) and \(\pi_i^{-1}(V)\)
are disjoint open sets in the product separating the two points.
Second countable: if \(\mathcal{B}_1\) and \(\mathcal{B}_2\) are
countable bases for \(X_1\) and \(X_2\), then
\(\mathcal{B} = \{B_1 \times B_2 : B_1 \in \mathcal{B}_1,\; B_2 \in \mathcal{B}_2\}\)
is countable (a countable product of countable sets is countable).
We claim \(\mathcal{B}\) is a basis for the product topology.
Every product basis element \(U_1 \times U_2\) (with \(U_i \in \tau_i\))
expands as \(U_1 \times U_2 = \bigcup_{\alpha, \beta} (B_{1,\alpha} \times B_{2,\beta})\),
where \(U_i = \bigcup_\alpha B_{i,\alpha}\) is the expansion of each factor in its own basis.
Since every open set in the product topology is a union of product basis elements,
it is also a union of elements of \(\mathcal{B}\) (by expanding each product
basis element as a union of products of basis elements of the factors).
Hence \(\mathcal{B}\) is a countable basis for the product topology.
The general case of \(n\) factors follows by induction: the product topology
on \(X_1 \times \cdots \times X_n\) agrees with the product topology on
\((X_1 \times \cdots \times X_{n-1}) \times X_n\) (a direct consequence of
the universal property, applied to both factorizations), so the two-factor
result applied at each step propagates Hausdorff and second countability
through all finite products.
Infinite Products and Tychonoff's Theorem
Definition: Product Topology (General Case)
Let \(\{X_\alpha\}_{\alpha \in A}\) be a family of topological spaces indexed
by an arbitrary (possibly infinite or uncountable) set \(A\). The
product topology on \(\prod_{\alpha \in A} X_\alpha\) is the
coarsest topology making every projection \(\pi_\alpha\) continuous.
Equivalently, a basis consists of sets of the form
\(\prod_{\alpha \in A} U_\alpha\) where \(U_\alpha \in \tau_\alpha\) for each
\(\alpha\) and \(U_\alpha = X_\alpha\) for all but finitely many indices.
The two descriptions agree by the following observation. For \(\pi_\alpha\) to be
continuous, the topology must contain every preimage
\(\pi_\alpha^{-1}(U_\alpha) = \{x : x_\alpha \in U_\alpha\}\) for \(U_\alpha \in \tau_\alpha\) —
that is, every "cylinder" constraining a single coordinate. The coarsest topology
containing all such cylinders consists of arbitrary unions of finite intersections
of cylinders, and finite intersections of cylinders are exactly the sets
\(\prod_\alpha U_\alpha\) with finitely many non-\(X_\alpha\) factors. The "finite"
restriction is a consequence of this construction, not an arbitrary choice: only
finite intersections are forced into the topology by the continuity requirements.
The "finitely many" restriction in the basis is precisely what makes infinite
products tractable — the decisive theorem for infinite products, stated next,
is one of the deepest results in general topology.
Theorem: Tychonoff's Theorem
An arbitrary product \(\prod_{\alpha \in A} X_\alpha\) of compact
topological spaces is compact in the product topology.
Tychonoff's theorem is logically equivalent to the Axiom of Choice. A standard
proof via Zorn's lemma or ultrafilters is treated in Munkres's topology text,
among other standard graduate references; the proof lies outside our scope,
but we have already seen its power: the proof
of the Banach-Alaoglu theorem
embedded the closed unit ball of \(\mathcal{X}^*\) into \(\prod_{x \in \mathcal{X}} [-\|x\|, \|x\|]\) — a product of
compact intervals — and invoked Tychonoff to conclude compactness. With the product topology now rigorously defined,
that argument stands on solid ground.
Example: Finite Products of Metric Spaces
If \((X_1, d_1)\) and \((X_2, d_2)\) are metric spaces, the product
topology on \(X_1 \times X_2\) is metrizable. For instance, the metric
\[
d\bigl((x_1, x_2),\, (y_1, y_2)\bigr)
= \max\{d_1(x_1, y_1),\, d_2(x_2, y_2)\}
\]
induces the product topology. (The metrics
\(d_1 + d_2\) and \(\sqrt{d_1^2 + d_2^2}\) also work — they all
induce the same product topology.) This confirms that our metric-space
intuition for \(\mathbb{R}^n = \mathbb{R} \times \cdots \times \mathbb{R}\)
is consistent with the product topology.
Connection to Parameter Spaces in Deep Learning
In multi-component models, the parameter space is naturally a product.
For a robotic arm with \(n\) joints, each joint's configuration lives
in a copy of \(S^1\) (the unit circle, parametrizing rotations)
or \(\mathbb{R}\) (translations), and the full configuration space
is a product \(S^1 \times \cdots \times S^1 \times \mathbb{R}^k\).
The product topology ensures that "small changes in each joint"
correspond to "small changes in the overall configuration" —
the correct notion of continuity for control algorithms and for
learning policies over configuration spaces.
Quotient Topology
The subspace and product constructions build larger spaces from
existing pieces. The quotient construction goes in the opposite
direction: it collapses parts of a space together,
identifying points according to an equivalence relation. This is
how circles, tori, projective spaces, and Lie group quotients are
formed — it is the most powerful and the most dangerous of the three
constructions, because the resulting space can lose properties
(like Hausdorff separation) that the original space possessed.
Definition: Quotient Topology
Let \((X, \tau)\) be a topological space and \(\sim\) an equivalence
relation on \(X\). Let \(X/{\sim}\) denote the set of equivalence
classes, and let \(\pi : X \to X/{\sim}\) be the quotient map
sending each point to its equivalence class: \(\pi(x) = [x]\).
The quotient topology on \(X/{\sim}\) is
\[
\tau_{X/\sim} = \{ V \subseteq X/{\sim} : \pi^{-1}(V) \in \tau \}.
\]
A set of equivalence classes is open if and only if the union of
those classes is open in \(X\).
Verification:
We check the three axioms, using that preimages commute with arbitrary unions
and intersections. (1) \(\pi^{-1}(\emptyset) = \emptyset \in \tau\) and
\(\pi^{-1}(X/{\sim}) = X \in \tau\), so \(\emptyset, X/{\sim} \in \tau_{X/\sim}\).
(2) If \(\{V_\alpha\} \subseteq \tau_{X/\sim}\), then
\(\pi^{-1}(\bigcup_\alpha V_\alpha) = \bigcup_\alpha \pi^{-1}(V_\alpha) \in \tau\)
by axiom 2 for \(\tau\), so \(\bigcup_\alpha V_\alpha \in \tau_{X/\sim}\).
(3) If \(V_1, \ldots, V_n \in \tau_{X/\sim}\), then
\(\pi^{-1}(\bigcap_{k=1}^n V_k) = \bigcap_{k=1}^n \pi^{-1}(V_k) \in \tau\)
by axiom 3 for \(\tau\), so \(\bigcap_{k=1}^n V_k \in \tau_{X/\sim}\).
By construction, the quotient map \(\pi\) is continuous and surjective.
The quotient topology is the finest topology on \(X/{\sim}\)
making \(\pi\) continuous — it declares a set open unless doing so would
force \(\pi\) to be discontinuous. This characterization, together with a
"passing to the quotient" criterion for continuity of maps out of \(X/{\sim}\),
forms the universal property that governs essentially every quotient
construction we will encounter.
Theorem: Universal Property of the Quotient Topology
Let \((X, \tau)\) be a topological space, \(\sim\) an equivalence relation
on \(X\), and \(\pi : X \to X/{\sim}\) the quotient map, with
\(X/{\sim}\) carrying the quotient topology \(\tau_{X/\sim}\).
- \(\pi\) is continuous, and \(\tau_{X/\sim}\) is the finest topology on
\(X/{\sim}\) making \(\pi\) continuous.
- For any topological space \(Y\) and any function \(f : X/{\sim} \to Y\),
\(f\) is continuous if and only if \(f \circ \pi : X \to Y\) is continuous.
Proof:
(1) For any \(V \in \tau_{X/\sim}\), \(\pi^{-1}(V) \in \tau\) by the
definition of \(\tau_{X/\sim}\), so \(\pi\) is continuous. To see
\(\tau_{X/\sim}\) is the finest such topology, let \(\tau'\) be any topology
on \(X/{\sim}\) making \(\pi\) continuous. Then for every \(V \in \tau'\),
\(\pi^{-1}(V) \in \tau\); by the definition of \(\tau_{X/\sim}\), this forces
\(V \in \tau_{X/\sim}\). Hence \(\tau' \subseteq \tau_{X/\sim}\), i.e.,
\(\tau_{X/\sim}\) contains every such \(\tau'\).
(2) (\(\Rightarrow\)) If \(f\) is continuous, then \(f \circ \pi\) is a
composition of continuous maps, hence continuous.
(\(\Leftarrow\)) Suppose \(f \circ \pi\) is continuous. For any open
\(U \subseteq Y\),
\[
\pi^{-1}(f^{-1}(U)) = (f \circ \pi)^{-1}(U) \in \tau,
\]
so \(f^{-1}(U) \in \tau_{X/\sim}\) by the definition of the quotient topology.
Hence \(f\) is continuous.
Key Examples
The following examples illustrate how familiar geometric objects arise
as quotient spaces. Each will reappear when we study smooth manifolds.
Example: The Circle \(S^1\)
Take the interval \([0, 1]\) and identify the endpoints: \(0 \sim 1\).
The quotient space \([0, 1]/{\sim}\) is homeomorphic to the
circle \(S^1\). The map \(\pi : [0, 1] \to S^1\) defined by
\(\pi(t) = e^{2\pi i t}\) is continuous, surjective, and respects \(\sim\)
(since \(\pi(0) = 1 = \pi(1)\)), so by the universal property it descends to a
continuous map \(\tilde{\pi}: [0,1]/{\sim} \to S^1\). Moreover,
\(\pi(s) = \pi(t)\) holds if and only if \(s - t \in \mathbb{Z}\); within \([0,1]\)
this forces \(s = t\) or \(\{s, t\} = \{0, 1\}\), which is exactly \(s \sim t\).
Hence \(\tilde{\pi}\) is injective, and combined with surjectivity of \(\pi\),
a bijection. Since \([0,1]/{\sim}\) is compact (as the image of the compact set
\([0,1]\) under the continuous quotient map) and \(S^1\) is Hausdorff,
\(\tilde{\pi}\) is a homeomorphism: a continuous bijection from a compact space
to a Hausdorff space sends closed sets to closed sets (closed subsets of a compact
space are compact, continuous images of compact sets are compact — both proved in
Compactness — and
compact subsets of Hausdorff spaces are closed),
hence is a homeomorphism by the
Homeomorphism Characterization via Open/Closed Maps.
Example: The Torus \(T^2\)
Define an equivalence relation on \(\mathbb{R}^2\) by
\((x, y) \sim (x + m, y + n)\) for all \((m, n) \in \mathbb{Z}^2\).
The quotient space \(\mathbb{R}^2 / \mathbb{Z}^2\) is the
torus \(T^2\). It is homeomorphic to the product
\(S^1 \times S^1\) via the map induced by
\((x, y) \mapsto (e^{2\pi i x}, e^{2\pi i y})\): this map is continuous,
surjective, and identifies two points precisely when they differ by an
element of \(\mathbb{Z}^2\), so it descends to a continuous bijection
\(\mathbb{R}^2 / \mathbb{Z}^2 \to S^1 \times S^1\). The formal verification
that this bijection is a homeomorphism parallels the \(S^1\) argument
above: every equivalence class in \(\mathbb{R}^2 / \mathbb{Z}^2\) has a
representative in the compact square \([0,1]^2\), so \(\mathbb{R}^2 / \mathbb{Z}^2\)
is compact, and \(S^1 \times S^1\) is Hausdorff (as a product of Hausdorff
spaces). Equivalently, take the unit square \([0,1]^2\) and glue opposite
edges: left to right and top to bottom.
Example: Real Projective Space \(\mathbb{R}P^n\)
Define an equivalence relation on the sphere \(S^n\) (carrying the subspace
topology inherited from \(\mathbb{R}^{n+1}\)) by \(x \sim -x\) (identifying
antipodal points). The quotient space \(S^n / {\sim}\) is the
real projective space \(\mathbb{R}P^n\). It parametrizes
lines through the origin in \(\mathbb{R}^{n+1}\): each line intersects
\(S^n\) in exactly two antipodal points, and the quotient collapses each
pair to a single point.
In contrast to the line with two origins below, \(\mathbb{R}P^n\) is Hausdorff:
the antipodal map \(a : S^n \to S^n,\; a(x) = -x\), is a homeomorphism, and the
equivalence relation \(x \sim -x\) is exactly the orbit relation of the
finite group \(\{\mathrm{id}, a\}\) acting freely on the compact Hausdorff
space \(S^n\). This is a clean instance of the proper group action
framework that preserves Hausdorff separation, to be developed formally in
the Lie group and manifold chapters.
The Hausdorff Warning
Unlike the subspace and product constructions, the quotient construction
does not automatically preserve the Hausdorff property:
the line with two origins is a quotient of two copies of \(\mathbb{R}\) (both Hausdorff) that fails to be Hausdorff.
This is not merely a curiosity — it is a genuine obstacle in manifold theory.
We now give the formal verification, using the quotient topology as defined above.
Theorem: The Line with Two Origins is Not Hausdorff
Let \(Y = (\mathbb{R} \times \{a\}) \sqcup (\mathbb{R} \times \{b\})\) be the
disjoint union of two copies of \(\mathbb{R}\), equipped with the
disjoint union topology (also called the coproduct
topology): a set \(W \subseteq Y\) is open if and only if its
restriction to each copy is open in that copy (each copy carrying the standard
topology of \(\mathbb{R}\)). Equivalently, each copy is itself open in \(Y\),
and \(Y\)'s open sets are unions of open subsets drawn from the two copies.
Define \(\sim\) on \(Y\) by \((x, a) \sim (x, b)\) for all \(x \neq 0\), with
each point equivalent to itself. Let \(L = Y / {\sim}\) with the quotient topology,
and let \(q : Y \to L\) denote the quotient map. Write \(0_a = q((0, a))\) and
\(0_b = q((0, b))\); these are distinct points of \(L\). Then \(L\) is not Hausdorff:
the points \(0_a\) and \(0_b\) cannot be separated by disjoint open sets.
Proof:
Suppose for contradiction that \(U, V \subseteq L\) are disjoint open sets
with \(0_a \in U\) and \(0_b \in V\). By the
quotient topology,
\(q^{-1}(U)\) and \(q^{-1}(V)\) are open in \(Y\).
Since \((0, a) \in q^{-1}(U)\) and the \(a\)-copy \(\mathbb{R} \times \{a\}\)
is open in \(Y\) with the topology transported from \(\mathbb{R}\), there
exists \(\epsilon > 0\) with \((-\epsilon, \epsilon) \times \{a\} \subseteq q^{-1}(U)\).
Similarly, there exists \(\delta > 0\) with
\((-\delta, \delta) \times \{b\} \subseteq q^{-1}(V)\).
Set \(\eta = \min(\epsilon, \delta) > 0\) and pick any \(x\) with \(0 < |x| < \eta\).
Then \((x, a) \in q^{-1}(U)\) and \((x, b) \in q^{-1}(V)\), so
\(q((x, a)) \in U\) and \(q((x, b)) \in V\).
But \(x \neq 0\) implies \((x, a) \sim (x, b)\), so
\(q((x, a)) = q((x, b))\). Hence \(U \cap V \ni q((x, a))\), contradicting
\(U \cap V = \emptyset\).
The lesson is concrete: after any quotient construction, Hausdorff
separation must be verified, not assumed.
Connection to Symmetry in Geometric Deep Learning
In Geometric Deep Learning, a neural network that is equivariant
under a group \(G\) effectively operates on the quotient space of inputs modulo
the \(G\)-action. For downstream topological and geometric analysis — continuity
of the learned feature map, convergence of training dynamics, or generalization
bounds in terms of the intrinsic geometry of the orbit space — this orbit space
is typically required to be Hausdorff. Heuristically, a non-Hausdorff quotient
allows "equivalent" and "inequivalent" inputs to be topologically indistinguishable,
undermining the notion of an invariant representation.
Rigorously, when the acting group is compact (as for \(\mathrm{SO}(3)\) on
\(\mathbb{R}^3\) in rotation-equivariant networks for 3D point clouds), the
orbit space is automatically Hausdorff — a consequence of the theory of
proper group actions.