Bases of Precompact Coordinate Balls
Having defined a topological manifold in the previous page, we now turn to
the technical properties that distinguish manifolds from arbitrary topological
spaces. The three defining conditions — Hausdorff, second-countable, and
locally Euclidean — are deceptively simple. Their real power comes from the
consequences they jointly imply, which form the infrastructure on which the
rest of manifold theory rests. In the previous page we noted that
second-countability
appears at first as a purely technical size constraint, with no obvious
geometric content. This page is where that condition begins to pay off. The
foundational result below shows that the topology of a manifold admits a
basis with two extremely strong properties: the basis elements are
coordinate balls, and each one has compact closure.
Recall that a subset \(A\) of a topological space \(X\) is said to be
precompact in \(X\) (synonymously, relatively compact) if
its closure \(\overline{A}\) in \(X\) is compact. The qualifier "in \(X\)"
matters: a set may be precompact when viewed inside one space and fail to be
so inside a larger one, since the closure operation depends on the ambient
space. The lemma below produces a basis in which every basis element is
precompact in the manifold \(M\) itself, not merely in some local chart
domain. This distinction will be essential in the proof.
Lemma: Basis of Precompact Coordinate Balls
Every topological \(n\)-manifold has a countable
basis
consisting of precompact
coordinate balls.
The proof is in two stages. We first establish the conclusion for manifolds
that happen to be covered by a single chart, where the problem reduces to a
construction in \(\mathbb{R}^n\). We then deduce the general case by
covering an arbitrary manifold with countably many such single-chart pieces
and assembling their bases.
Proof:
Let \(M\) be a topological \(n\)-manifold.
Step 1 — Single-chart case:
Suppose first that \(M\) admits a global coordinate chart, that is, a
homeomorphism \(\varphi : M \to \widehat{M} \subseteq \mathbb{R}^n\) onto
an open subset \(\widehat{M}\) of Euclidean space. Let \(\mathcal{B}\)
be the collection of all open balls \(B_r(x) \subseteq \mathbb{R}^n\)
such that \(r\) is rational, \(x\) has rational coordinates, and
\(\overline{B_{r'}(x)} \subseteq \widehat{M}\) for some rational
\(r' > r\). Each such ball is contained in \(\widehat{M}\) together
with a slightly larger closed ball, so its closure in \(\mathbb{R}^n\) —
which by
the Heine–Borel theorem
is compact — is contained in \(\widehat{M}\). Since compactness is
intrinsic (a property of a space, independent of any ambient containing it),
this closed set remains compact when regarded as a subset of \(\widehat{M}\),
and it agrees with the closure of \(B\) in \(\widehat{M}\) because it
already lies in \(\widehat{M}\). Therefore each \(B \in \mathcal{B}\)
is precompact in \(\widehat{M}\).
The collection \(\mathcal{B}\) is countable, since it is indexed by
pairs of rationals. It is a basis for the topology of \(\widehat{M}\):
given any open \(W \subseteq \widehat{M}\) and any point \(y \in W\),
choose \(\rho > 0\) with \(\overline{B_\rho(y)} \subseteq W\), then pick
a rational \(x\) close to \(y\) and rationals \(r' > r\) small enough
that \(y \in B_r(x)\) and \(\overline{B_{r'}(x)} \subseteq B_\rho(y)
\subseteq \widehat{M}\); the resulting ball lies in \(\mathcal{B}\) and
contains \(y\) inside \(W\).
Pulling back through the homeomorphism, the collection
\[
\left\{ \varphi^{-1}(B) : B \in \mathcal{B} \right\}
\]
is a countable basis for the topology of \(M\). Each set
\(\varphi^{-1}(B)\) is a coordinate ball, with the restriction
\(\varphi|_{\varphi^{-1}(B)}\) as its coordinate map. It remains to
verify that each \(\varphi^{-1}(B)\) is precompact in \(M\). The
closure of \(B\) in \(\widehat{M}\) is compact, and the
continuous image of a compact set
under \(\varphi^{-1}\) is compact in \(M\). Because \(M\) is Hausdorff
(as a manifold),
this compact set is closed in \(M\).
It is therefore a closed set containing \(\varphi^{-1}(B)\), so the
closure of \(\varphi^{-1}(B)\) in \(M\) is contained in it. A closed
subset of a compact set is compact, hence \(\varphi^{-1}(B)\) is
precompact in \(M\).
Step 2 — General case:
Now let \(M\) be an arbitrary topological \(n\)-manifold. By definition,
each point of \(M\) lies in the domain of some chart. The collection of
all chart domains forms an open cover of \(M\). Since \(M\) is
second-countable,
every open cover admits a countable subcover,
so \(M\) is covered by
countably many charts \(\{(U_i, \varphi_i)\}_{i \in \mathbb{N}}\).
Each coordinate domain \(U_i\), being an open subset of \(M\), is itself a
topological \(n\)-manifold
covered by the single chart \((U_i, \varphi_i)\). Step 1 therefore
produces, for each \(i\), a countable basis \(\mathcal{B}_i\) for the
topology of \(U_i\) consisting of coordinate balls that are precompact
in \(U_i\). The union
\[
\mathcal{B} = \bigcup_{i \in \mathbb{N}} \mathcal{B}_i
\]
is a countable collection of open subsets of \(M\); since each
\(\mathcal{B}_i\) is a basis for \(U_i\) and the \(U_i\) cover \(M\),
the collection \(\mathcal{B}\) is a basis for the topology of \(M\).
It remains only to upgrade precompactness from "in \(U_i\)" to "in \(M\)."
Let \(V \in \mathcal{B}_i\), so the closure of \(V\) in \(U_i\) — call
it \(\overline{V}^{U_i}\) — is compact. Since \(M\) is Hausdorff,
this compact subset is also closed in \(M\).
The closure of \(V\) in \(M\), denoted \(\overline{V}^M\), is the
smallest closed subset of \(M\) containing \(V\); since
\(\overline{V}^{U_i}\) is itself a closed set in \(M\) containing \(V\),
we have \(\overline{V}^M \subseteq \overline{V}^{U_i}\). A closed subset
of a compact set is compact, so \(\overline{V}^M\) is compact, and
\(V\) is precompact in \(M\). This holds for every \(V \in \mathcal{B}\),
completing the proof. \(\blacksquare\)
Both technical hypotheses on a manifold contribute to this single lemma:
second-countability gives countability of the basis, and Hausdorffness
gives the precompactness upgrade from a chart domain to the ambient
manifold. The lemma is the workhorse of the present page. Local
compactness, treated next, is a one-line corollary. The connectivity
properties of manifolds rely on the fact that the basis consists of
coordinate balls — geometrically simple sets that are path-connected by
construction. And the paracompactness theorem, the technical climax of
the page, builds an exhaustion of \(M\) by compact sets out of the
precompact coordinate balls supplied here.
Connectivity of Manifolds
The existence of a basis of coordinate balls has immediate consequences
for the connectivity properties of manifolds. The basic facts of
connectivity in topological spaces — that a space may be
connected
in the sense of not splitting into two disjoint nonempty open sets, or
path-connected
in the sense that any two points can be joined by a continuous path —
were established for metric spaces in earlier work. The two notions are
distinct in general: there exist connected spaces that fail to be
path-connected, the classical example being the topologist's sine curve.
For manifolds, however, the two notions coincide, and we will see this
coincidence is forced by purely local Euclidean structure.
The bridge between the two notions is a local one: it suffices that each
point have arbitrarily small path-connected neighborhoods. A topological
space \(X\) is locally path-connected if it admits a basis of
path-connected open sets. For manifolds, the basis of coordinate balls
supplied by the previous section is automatically a basis of
path-connected sets: each coordinate ball is homeomorphic to an open ball
in \(\mathbb{R}^n\), and open balls are path-connected because any two
points in an open ball are joined by the straight-line segment between
them, which lies entirely within the ball. This single observation drives
all four statements below.
Proposition: Connectivity of Manifolds
Let \(M\) be a topological manifold. Then:
-
\(M\) is locally path-connected.
-
\(M\) is connected if and only if it is path-connected.
-
The
connected components
of \(M\) coincide with its path components.
-
\(M\) has at most countably many connected components, each of
which is an open subset of \(M\) and a connected topological
manifold of the same dimension as \(M\).
Proof:
(a):
By the previous section, \(M\) admits a basis of coordinate balls.
Each coordinate ball is homeomorphic to an open ball in
\(\mathbb{R}^n\). Open balls in \(\mathbb{R}^n\) are path-connected:
for any two points \(x, y\) in an open ball \(B_r(z)\), the segment
\(t \mapsto (1-t)x + ty\) for \(t \in [0,1]\) is a continuous path
from \(x\) to \(y\), and convexity of the ball ensures the segment
stays inside. Path-connectedness transfers under homeomorphism, so
each coordinate ball is path-connected. Therefore \(M\) admits a
basis of path-connected open sets, which is the definition of local
path-connectedness.
(b) and (c):
It is a standard fact of general topology that in a locally
path-connected space, path components are open, and a space is
connected if and only if it is path-connected. In any topological
space \(X\), define the path component of a point \(p\) to be the set
of all points reachable from \(p\) by a continuous path. Path
components partition \(X\) into equivalence classes. In a locally
path-connected space, each path component is open: given \(p\) in the
path component \(P\), local path-connectedness gives a path-connected
open neighborhood \(U\) of \(p\), and since every point of \(U\) is
reachable from \(p\) by a path, \(U\) lies entirely in \(P\). The path
components are therefore disjoint open subsets covering \(X\). If
\(X\) had more than one path component, say \(P\) and its complement
\(X \setminus P\) (which is the union of the remaining path components
and hence open), then \(X\) would decompose as the disjoint union of
two nonempty open sets — i.e., \(X\) would be disconnected. Thus a
connected, locally path-connected space has a single path component —
it is path-connected. Conversely,
every path-connected space is connected.
For (c), let \(C\) be a connected component of \(X\) and let \(P\) be
a path component intersecting \(C\). Path-connectedness implies
connectedness, so \(P\) is connected; since \(C\) is a maximal
connected subset and \(P \cap C\) is nonempty, we have \(P \subseteq
C\). Suppose \(C\) properly contains \(P\). Then \(C = P \cup (C
\setminus P)\), where \(C \setminus P\) is the union of the other path
components meeting \(C\), which is open by the same argument as before.
The two pieces are nonempty, disjoint, and open in \(C\), contradicting
the connectedness of \(C\). Hence \(C = P\). Applying these facts to
\(M\), which is locally path-connected by part (a), yields (b) and (c).
(d):
By (c), the connected components of \(M\) are the same as the path
components, and the argument above shows path components in a
locally path-connected space are open. The connected components of
\(M\) are therefore an open cover of \(M\) by mutually disjoint sets.
Since \(M\) is second-countable, this cover admits a
countable subcover;
but disjoint nonempty open sets cannot be discarded from a
cover, so the cover itself must be countable. Hence \(M\) has at
most countably many connected components.
Each connected component is an open subset of \(M\), and by
the open submanifold property,
every open subset of a topological \(n\)-manifold is itself a
topological \(n\)-manifold under the subspace topology. Each
component is therefore a topological manifold of the same dimension
as \(M\), and it is connected by definition. \(\blacksquare\)
The principal content of the proposition is the equivalence of connectedness
and path-connectedness for manifolds — a feature manifolds inherit from
Euclidean space at the local level and which holds globally because
coordinate balls knit together coherently. In practice, this means that
when treating a connected manifold, the apparently weaker hypothesis of
connectedness gives access to the stronger conclusion that any two points
can be joined by a continuous path — a frequently useful tool, for
instance in propagating local information along curves.
Part (d) carries an additional structural consequence. A topological
manifold decomposes canonically into countably many connected components,
each of which is itself a manifold of the same dimension. This
decomposition is in effect a reduction: any question about a general
\(n\)-manifold can, if convenient, be reduced to the same question on
each connected component separately. For this reason it is common in the
literature, and in subsequent pages, to assume implicitly that a
manifold is connected when no generality is lost.
Countability of the Fundamental Group
The basis of coordinate balls has one further consequence, of a different
character from the point-set properties established so far. It concerns the
fundamental group,
the algebraic invariant that records how loops in a space fail to contract.
For a manifold this group cannot be arbitrarily large: it is always countable.
The result will matter later, when the relationship between a manifold and its
covering spaces is taken up, where countability of the fundamental group
bounds the supply of covers.
Proposition: The Fundamental Group of a Manifold Is Countable
Let \(M\) be a topological manifold. Then for every \(p \in M\), the
fundamental group \(\pi_1(M, p)\) is countable.
Proof:
By the
basis of precompact coordinate balls,
there is a countable collection \(\mathcal{B}\) of coordinate balls
covering \(M\). Each ball \(B \in \mathcal{B}\) is homeomorphic to an open
ball in \(\mathbb{R}^n\), hence path-connected; and for any pair
\(B, B' \in \mathcal{B}\), the intersection \(B \cap B'\) has at most
countably many components, each open and therefore path-connected by local
path-connectedness. The bound on the number of components is itself a
consequence of second-countability, exactly as for the components of \(M\)
in the preceding section. From each component of each intersection
\(B \cap B'\) — including the case \(B = B'\), whose single component
is \(B\) itself — choose one point, and let \(X\) be the resulting
set. As a countable union of finite or countable selections indexed by the
countable set \(\mathcal{B} \times \mathcal{B}\), the set \(X\) is countable.
For each \(B \in \mathcal{B}\) and each pair \(x, x' \in X \cap B\), fix
once and for all a path \(h^{B}_{x, x'}\) from \(x\) to \(x'\) lying in
\(B\); this is possible because \(B\) is path-connected. Call a loop based
at a chosen base point a special loop if it equals a finite
product of paths of the form \(h^{B}_{x, x'}\). The set of special loops is
countable: each is determined by a finite sequence drawn from the countable
family of available paths \(h^{B}_{x, x'}\), and the set of finite sequences
from a countable set is countable.
Because the fundamental groups based at any two points of the same component
of \(M\) are isomorphic — an isomorphism furnished by conjugation along
a connecting path, as established when the
fundamental group
was introduced — and \(X\) meets every component of \(M\), it costs no
generality to take the base point \(p\) to lie in \(X\). Each special loop
then determines an element of \(\pi_1(M, p)\), and there are only countably
many special loops; the proposition therefore follows once we show that
every element of \(\pi_1(M, p)\) is represented by a special loop.
Let \(f : [0, 1] \to M\) be a loop based at \(p\). The preimages \(f^{-1}(B)\),
as \(B\) ranges over \(\mathcal{B}\), form an open cover of the unit interval;
since \([0, 1]\) is compact, the cover has a finite subcover, and a
Lebesgue-number argument yields a
partition \(0 = a_0 < a_1 < \cdots < a_k = 1\) such that \(f\) carries each
subinterval \([a_{i-1}, a_i]\) into a single ball \(B_i \in \mathcal{B}\).
Writing \(f_i\) for the restriction of \(f\) to \([a_{i-1}, a_i]\),
reparametrized to the domain \([0, 1]\), the loop factors as a
product of paths,
\(f = f_1 \cdot f_2 \cdots f_k\).
At each interior division point \(f(a_i) \in B_i \cap B_{i+1}\); choose
\(x_i \in X\) lying in the same component of \(B_i \cap B_{i+1}\) as
\(f(a_i)\), and let \(g_i\) be a path in that component from \(x_i\) to
\(f(a_i)\), with the convention \(x_0 = x_k = p\) and \(g_0, g_k\) constant.
Splicing in each \(g_i\) together with its reverse \(\bar{g}_i\) — whose
composite \(g_i \cdot \bar{g}_i\) is
path-homotopic
to a constant path, by the same retracing that inverts a loop — leaves
the path class of \(f\) unchanged and regroups it as
\[
f \;\sim\; \tilde{f}_1 \cdot \tilde{f}_2 \cdots \tilde{f}_k,
\qquad
\tilde{f}_i = g_{i-1} \cdot f_i \cdot \bar{g}_i,
\]
where each \(\tilde{f}_i\) is a path in \(B_i\) running from \(x_{i-1}\) to
\(x_i\): the path \(g_{i-1}\) starts at \(x_{i-1}\) and the reverse
\(\bar{g}_i\) ends at \(x_i\), so the endpoints chain
\(x_{i-1} \to f(a_{i-1}) \to f(a_i) \to x_i\). Since \(B_i\), being a
coordinate ball, is
simply connected,
any two paths in \(B_i\) with the same endpoints are path-homotopic; in
particular \(\tilde{f}_i \sim h^{B_i}_{x_{i-1}, x_i}\). Therefore \(f\) is
path-homotopic to the special loop
\(h^{B_1}_{x_0, x_1} \cdots h^{B_k}_{x_{k-1}, x_k}\), and its class is one of
the countably many classes carried by special loops. \(\blacksquare\)
The countability is a genuine restriction. The fundamental group of a manifold
may well be infinite — the circle already has fundamental group
\(\mathbb{Z}\) — but it can never attain the cardinality of, say, the
continuum. The proof extracts this bound from nothing more than the countable
basis: a loop is cut into finitely many arcs, each confined to a coordinate ball,
and the countable bookkeeping of where consecutive arcs meet suffices to pin its
class to a countable list. The same mechanism, with coordinate half-balls in
place of balls, carries the result to manifolds with boundary without change.
Local Compactness
The basis of precompact coordinate balls established in the first section
has another immediate consequence, this time concerning local behavior
rather than global connectivity. A topological space \(X\) is said to be
locally compact if every point of \(X\) has a neighborhood
contained in a compact subset of \(X\). For Hausdorff spaces — the setting
that concerns us, since manifolds are Hausdorff by definition — this is
equivalent to either of the more concrete conditions that every point have
a precompact neighborhood, or that \(X\) have a basis of precompact open
subsets. We will use these formulations interchangeably. Local compactness
is a mild form of finiteness: it does not require the space itself to be
compact, but it does require that one can always retreat to a compact piece
around any given point.
Local compactness is the technical hypothesis under which many of the
classical tools of analysis transfer from Euclidean space to more general
settings. Integration against a measure of compact support, the existence
of compactly supported continuous functions, and Urysohn-type extension
results all rest on it. For manifolds, local compactness is essentially
automatic, and the proof reduces to a single appeal to the basis
constructed in the previous section.
Proposition: Manifolds Are Locally Compact
Every topological manifold is locally compact.
Proof:
Let \(M\) be a topological manifold and \(p \in M\). The
basis of precompact coordinate balls
established in the first section is, in particular, a basis for the
topology of \(M\), so there exists some basis element \(V\) with
\(p \in V\). By construction, \(V\) is a precompact open subset of
\(M\), meaning its closure in \(M\) is compact. Hence \(p\) has a
neighborhood with compact closure, which is the defining condition of
local compactness. Since \(p\) was arbitrary, \(M\) is locally
compact. \(\blacksquare\)
The proof reveals the structural role of the basis of precompact
coordinate balls. The statement of
local compactness requires only one precompact neighborhood at each point,
whereas that basis supplies an entire family of them. The surplus is real
content: it will be used in the next section, where the construction of a
locally finite refinement of an open cover relies on the freedom to choose
precompact basis elements sitting inside any prescribed open set.
Paracompactness
We arrive at the central result of this page. In the previous page on
topological manifolds, we noted that second-countability is included in the
definition primarily because of its downstream consequences — most
importantly, the existence of partitions of unity, the device by which
local constructions on coordinate charts are glued into global structures
on the manifold. The intermediate step between second-countability and
partitions of unity is the topological property called paracompactness,
which we now establish for manifolds.
The route from second-countability to paracompactness passes through every
earlier result of this page. The basis of precompact coordinate balls
(§1) produces an exhaustion of the manifold by compact sets. Local
compactness (§3) ensures that this exhaustion has the right local
properties. The compact exhaustion that organizes the
proof of paracompactness is, in turn, the technical content packed into
the seemingly innocuous condition that the topology of \(M\) be describable
by countably many open sets. This is the place where second-countability
most explicitly pays off.
Definitions
Paracompactness concerns the existence of well-behaved refinements of
arbitrary open covers. We begin with the three notions involved.
Definition: Locally Finite Collection
Let \(M\) be a topological space. A collection \(\mathcal{X}\) of
subsets of \(M\) is said to be locally finite if every
point of \(M\) has a
neighborhood
that intersects at most finitely
many sets in \(\mathcal{X}\).
Local finiteness is strictly weaker than finiteness: an infinite collection
can be locally finite if its members are spread out across \(M\) in such a
way that any single point sees only finitely many of them in its vicinity.
The collection of intervals \(\{(n - 1, n + 1) : n \in \mathbb{Z}\}\) in
\(\mathbb{R}\) is a typical example: it is infinite, yet around any point
of \(\mathbb{R}\) only two of the intervals overlap.
Definition: Refinement of a Cover
Let \(\mathcal{U}\) be a cover of a topological space \(M\). Another
cover \(\mathcal{V}\) of \(M\) is called a refinement
of \(\mathcal{U}\) if for each \(V \in \mathcal{V}\) there exists some
\(U \in \mathcal{U}\) with \(V \subseteq U\). The refinement is called
an open refinement if every member of \(\mathcal{V}\)
is open, and a locally finite refinement if
\(\mathcal{V}\) is locally finite as a collection of subsets of \(M\).
Informally, a refinement consists of smaller sets, each contained inside
one of the original cover elements; a locally finite refinement adds the
requirement that the smaller sets do not pile up at any point of \(M\).
Definition: Paracompact Space
A topological space \(M\) is called paracompact if
every open cover of \(M\) admits an open, locally finite refinement.
Paracompactness is best understood as a mild compactness-like condition:
every compact space is paracompact (a finite subcover is trivially a
locally finite refinement), but paracompactness allows the space to be
"infinitely large" provided that its open covers can always be tamed by
refinement.
A Technical Lemma on Locally Finite Collections
The closure operation interacts well with local finiteness: passing to
closures preserves the property, and closures commute with arbitrary
unions when the collection is locally finite. The second statement is
notable because, in general, the closure of an arbitrary union is only
contained in the union of closures, not equal to it.
Lemma: Closures of Locally Finite Collections
Let \(\mathcal{X}\) be a locally finite collection of subsets of a
topological space \(M\). Then:
-
The collection \(\{\overline{X} : X \in \mathcal{X}\}\) of closures
is also locally finite.
-
\(\displaystyle \overline{\bigcup_{X \in \mathcal{X}} X} = \bigcup_{X \in \mathcal{X}} \overline{X}\).
Proof:
(a):
Let \(p \in M\) and choose a neighborhood \(W\) of \(p\) that
intersects only finitely many members \(X_1, \ldots, X_k\) of
\(\mathcal{X}\). If \(X \in \mathcal{X}\) is distinct from all of
\(X_1, \ldots, X_k\), then \(W \cap X = \emptyset\). Since \(W\) is
open, this implies \(W \cap \overline{X} = \emptyset\). For if some
point \(q\) lay in both \(W\) and \(\overline{X}\), then \(W\) would
be an open neighborhood of \(q\), and since \(q \in \overline{X}\),
every open neighborhood of \(q\) meets \(X\); in particular
\(W \cap X \ne \emptyset\), contradicting \(W \cap X = \emptyset\).
Thus \(W\) intersects \(\overline{X}\) for at most the finitely many
indices \(X = X_1, \ldots, X_k\). Hence
\(\{\overline{X} : X \in \mathcal{X}\}\) is locally finite.
(b):
The inclusion
\(\bigcup_{X \in \mathcal{X}} \overline{X} \subseteq \overline{\bigcup_{X \in \mathcal{X}} X}\)
holds for any collection of sets and uses no hypothesis on
\(\mathcal{X}\). For the reverse inclusion, suppose \(p \in
\overline{\bigcup_{X \in \mathcal{X}} X}\). By local finiteness,
choose a neighborhood \(W\) of \(p\) intersecting only finitely many
members \(X_1, \ldots, X_k\). Then
\[
W \cap \bigcup_{X \in \mathcal{X}} X = W \cap (X_1 \cup \cdots \cup X_k).
\]
Since \(p \in \overline{\bigcup_{X \in \mathcal{X}} X}\), every open
neighborhood of \(p\) meets \(\bigcup_{X \in \mathcal{X}} X\). Given
any open neighborhood \(N\) of \(p\), the intersection \(N \cap W\) is
also an open neighborhood of \(p\) and is contained in \(W\); hence
\((N \cap W) \cap \bigcup_{X \in \mathcal{X}} X = (N \cap W) \cap (X_1 \cup \cdots \cup X_k)\)
is nonempty, which in particular implies \(N\) meets \(X_1 \cup \cdots \cup X_k\).
Therefore \(p \in \overline{X_1 \cup \cdots \cup X_k} = \overline{X_1} \cup \cdots \cup
\overline{X_k}\), where the final equality uses the fact that closure
of a finite union is the union of closures. Hence \(p \in \overline{X_i}\)
for some \(i\), so \(p \in \bigcup_{X \in \mathcal{X}} \overline{X}\).
\(\blacksquare\)
Compact Exhaustions
The proof of paracompactness for manifolds proceeds by organizing \(M\)
into an increasing sequence of compact pieces. We first establish the
relevant notion and its existence on manifolds.
Definition: Compact Exhaustion
A sequence \((K_i)_{i = 1}^\infty\) of compact subsets of a topological
space \(X\) is called an exhaustion of \(X\) by compact sets
if \(X = \bigcup_{i} K_i\) and \(K_i \subseteq \operatorname{int}(K_{i+1})\)
for every \(i\).
The crucial structural feature is the nesting condition
\(K_i \subseteq \operatorname{int}(K_{i+1})\): each compact piece is
contained not merely in the next, but in its interior, providing a
"buffer" of open space around \(K_i\) inside \(K_{i+1}\). This buffer
is what allows arguments by induction to extend constructions from one
compact piece to the next without boundary effects.
Lemma: Manifolds Admit Compact Exhaustions
Every topological manifold admits an exhaustion by compact sets.
Proof:
Let \(M\) be a topological manifold. By the
basis of precompact coordinate balls,
\(M\) has a countable basis \(\{B_i\}_{i = 1}^\infty\) of precompact
open subsets. By the
Lindelöf property,
the family \(\{B_i\}\) is itself a countable open cover of \(M\) (any
basis covers the space). We construct an exhaustion by induction.
Set \(K_1 = \overline{B_1}\), which is compact by precompactness of
\(B_1\). Suppose by induction that compact sets \(K_1, \ldots, K_k\)
have been constructed with \(K_{j-1} \subseteq \operatorname{int}(K_j)\)
for \(2 \le j \le k\), and \(B_j \subseteq K_j\) for \(1 \le j \le k\).
Since \(K_k\) is compact and the collection \(\{B_i\}\) covers \(M\),
finitely many of the \(B_i\) cover \(K_k\). Let \(m_k\) be an integer
at least as large as the largest index appearing among these chosen
sets, and also at least \(k + 1\); then
\(\{B_1, B_2, \ldots, B_{m_k}\}\) contains a finite cover of \(K_k\)
and includes \(B_{k+1}\). Define
\[
K_{k+1} = \overline{B_1} \cup \overline{B_2} \cup \cdots \cup \overline{B_{m_k}}.
\]
This is a finite union of compact sets, hence compact. The union
\(B_1 \cup \cdots \cup B_{m_k}\) is open and contains \(K_k\);
therefore \(K_k\) is contained in the interior of \(K_{k+1}\).
Moreover, \(B_{k+1} \subseteq K_{k+1}\) by construction.
The resulting sequence \((K_i)_{i=1}^\infty\) is increasing, each
\(K_i\) is compact, and \(K_i \subseteq \operatorname{int}(K_{i+1})\).
Since \(B_i \subseteq K_i\) for each \(i\) and the \(B_i\) cover \(M\),
we have \(M = \bigcup_i K_i\). Thus \((K_i)\) is the required
exhaustion. \(\blacksquare\)
With the compact exhaustion in hand, we are ready to state and prove the
main theorem of this page. The statement we give is slightly stronger than
plain paracompactness: it asserts that the locally finite refinement can
be chosen to consist of elements of any prescribed basis for the topology.
This strengthening is what enables applications such as the construction
of smooth partitions of unity, where one wants to refine an arbitrary
open cover into a cover by coordinate balls.
Theorem: Manifolds Are Paracompact
Every topological manifold is paracompact. More precisely, given a
topological manifold \(M\), an open cover \(\mathcal{U}\) of \(M\),
and any basis \(\mathcal{B}\) for the topology of \(M\), there exists
a countable, locally finite open refinement of \(\mathcal{U}\)
consisting of elements of \(\mathcal{B}\).
The proof, which builds the refinement from the compact exhaustion just
established, occupies the remainder of this section.
Proof of the Paracompactness Theorem
Let \(M\), \(\mathcal{U}\), and \(\mathcal{B}\) be as in the statement of
the theorem. By the previous subsection, \(M\) admits a compact
exhaustion \((K_j)_{j = 1}^\infty\), with \(K_j \subseteq
\operatorname{int}(K_{j+1})\) and \(M = \bigcup_j K_j\). It will be
convenient throughout the argument to set \(K_j = \emptyset\) for \(j \le 0\),
so that expressions like \(\operatorname{int}(K_{j+2}) \setminus K_{j-1}\)
are meaningful at all indices \(j \ge 1\).
For each integer \(j \ge 0\), define
\[
V_j = K_{j+1} \setminus \operatorname{int}(K_j), \qquad
W_j = \operatorname{int}(K_{j+2}) \setminus K_{j-1}.
\]
Two structural facts about these sets drive the entire argument. First,
\(V_j\) is compact: it is a closed subset of the compact set \(K_{j+1}\),
since \(K_j\) and hence \(\operatorname{int}(K_j)\) is contained in
\(K_{j+1}\), and the complement of an open set in a compact set is closed
and therefore compact. Second, \(V_j\) is contained in \(W_j\): a point
\(x \in V_j\) lies in \(K_{j+1} \subseteq \operatorname{int}(K_{j+2})\),
and it does not lie in \(\operatorname{int}(K_j)\), hence not in
\(K_{j-1}\) either, since \(K_{j-1} \subseteq \operatorname{int}(K_j)\).
The sets \(W_j\) are open: \(K_{j-1}\) is compact in the Hausdorff space
\(M\) and hence
closed,
so \(M \setminus K_{j-1}\) is open, and
\(W_j = \operatorname{int}(K_{j+2}) \cap (M \setminus K_{j-1})\) is an
intersection of two open sets.
The compact pieces \(V_j\) collectively cover \(M\): if \(p \in M\), let
\(i\) be the smallest positive integer with \(p \in K_i\) (which exists
because the \(K_i\) exhaust \(M\)), and set \(j = i - 1 \ge 0\). By
minimality, \(p \notin K_{i-1} = K_j\) (using the convention \(K_j =
\emptyset\) for \(j \le 0\)), so \(p \notin \operatorname{int}(K_j)\), and
consequently \(p \in K_{j+1} \setminus \operatorname{int}(K_j) = V_j\).
Thus \(M = \bigcup_{j \ge 0} V_j\). Each \(V_j\) is enclosed inside the
open "shell" \(W_j\), and the shells overlap only between nearby indices,
in a sense we now make precise.
We now construct the refinement. Fix \(j\). For each \(x \in V_j\), choose
some \(X_x \in \mathcal{U}\) containing \(x\), which exists because
\(\mathcal{U}\) covers \(M\). The set \(X_x \cap W_j\) is open and contains
\(x\), so by the basis property of \(\mathcal{B}\) there is some
\(B_x \in \mathcal{B}\) with
\[
x \in B_x \subseteq X_x \cap W_j.
\]
The family \(\{B_x : x \in V_j\}\) is an open cover of the compact set
\(V_j\), so it admits a finite subcover; call this finite subcover
\(\mathcal{V}_j \subseteq \mathcal{B}\). Each set in \(\mathcal{V}_j\) is
contained in some element of \(\mathcal{U}\) (namely the corresponding
\(X_x\)), and each set in \(\mathcal{V}_j\) is contained in \(W_j\).
Define
\[
\mathcal{V} = \bigcup_{j = 0}^\infty \mathcal{V}_j.
\]
The collection \(\mathcal{V}\) has each of the required properties. First,
it covers \(M\), since the \(V_j\) cover \(M\) and each \(\mathcal{V}_j\)
covers \(V_j\). Second, it is a countable union of finite collections,
hence countable. Third, every member of \(\mathcal{V}\) lies in \(\mathcal{B}\),
and every member is contained in some element of \(\mathcal{U}\), so
\(\mathcal{V}\) is a refinement of \(\mathcal{U}\) consisting of basis
elements.
It remains to verify local finiteness. The key observation is that the
shells \(W_j\) themselves overlap only between nearby indices. Suppose
\(W_j \cap W_{j'} \ne \emptyset\) with \(j \le j'\). Then \(W_j \subseteq
\operatorname{int}(K_{j+2}) \subseteq K_{j+2}\), and
\(W_{j'} \cap K_{j'-1} = \emptyset\); so a common point lies in
\(K_{j+2}\) and in the complement of \(K_{j'-1}\), which forces
\(K_{j+2} \not\subseteq K_{j'-1}\). Since the sequence \((K_i)\) is
increasing, \(A \le B\) implies \(K_A \subseteq K_B\); the contrapositive
of this implication, applied to \(A = j + 2\) and \(B = j' - 1\), yields
\(j + 2 > j' - 1\), which rearranges to \(j' \le j + 2\). Combined with
the assumption \(j \le j'\), this yields \(0 \le j' - j \le 2\). A
symmetric argument for \(j' \le j\) gives \(0 \le j - j' \le 2\). Hence
\(W_j \cap W_{j'} = \emptyset\) whenever \(|j - j'| \ge 3\).
Now let \(p \in M\). Since the \(K_j\) exhaust \(M\), there is some
\(j_0 \ge 0\) with \(p \in \operatorname{int}(K_{j_0 + 2})\), and
\(\operatorname{int}(K_{j_0 + 2})\) is an open neighborhood of \(p\).
This neighborhood is disjoint from \(W_{j'}\) for every
\(j' \ge j_0 + 3\): for such \(j'\),
\(j' - 1 \ge j_0 + 2\), so \(K_{j'-1} \supseteq K_{j_0 + 2} \supseteq
\operatorname{int}(K_{j_0 + 2})\), and \(W_{j'}\) lies in the complement
of \(K_{j'-1}\). The neighborhood \(\operatorname{int}(K_{j_0 + 2})\) of
\(p\) therefore meets at most the shells \(W_0, W_1, \ldots, W_{j_0 + 2}\) —
finitely many in total. Since each member of \(\mathcal{V}_{j'}\) is
contained in \(W_{j'}\), the neighborhood \(\operatorname{int}(K_{j_0 + 2})\)
meets only members of \(\mathcal{V}_0 \cup \mathcal{V}_1 \cup \cdots \cup
\mathcal{V}_{j_0 + 2}\), which is a finite union of finite families and
therefore finite. Hence \(p\) has a neighborhood intersecting only
finitely many members of \(\mathcal{V}\). Since \(p\) was arbitrary,
\(\mathcal{V}\) is locally finite. \(\blacksquare\)
Significance
The paracompactness theorem is the technical engine of much of what
follows in manifold theory. Its most important immediate consequence —
developed later in this series, once smooth maps are in hand — is the
existence of
smooth partitions of unity:
collections of smooth functions, subordinate to a given open cover, whose
supports form a locally finite collection and whose values sum to \(1\) at every point.
Partitions of unity convert local data on charts into global data on the
manifold, and this conversion is the mechanism behind virtually every
global construction in differential geometry: smooth bump functions,
smooth approximation theorems, Riemannian metrics on arbitrary smooth
manifolds, the integration of differential forms, and many more. The
strengthening that \(\mathcal{V}\) consists of basis elements is what
permits partitions of unity to be built from a basis of smooth coordinate
balls rather than from an arbitrary open cover.