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
precompact
in \(X\) when 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
\(r' \gt r\). Each \(B = B_r(x) \in \mathcal{B}\) is contained in
\(\widehat{M}\) together with a slightly larger closed ball, so its
closure in \(\mathbb{R}^n\) is contained in \(\widehat{M}\), and
that closure is compact by
the Heine-Borel theorem.
Since compactness is intrinsic (a property of a space, independent of
any ambient containing it), that closure 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 its members are
indexed by a rational center and a rational radius. It is a basis for
the topology of \(\widehat{M}\).
Given any open \(W \subseteq \widehat{M}\) and any point \(y \in W\),
choose \(\rho \gt 0\) with \(\overline{B_\rho(y)} \subseteq W\), then
pick a rational \(x\) close to \(y\) and rationals \(r' \gt r\) 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,
this 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 \(V \in \mathcal{B}_i\), and write \(\overline{V}^{U_i}\)
for the closure of \(V\) in \(U_i\), which 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.
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 were established for metric spaces in earlier work. 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. 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, which is what it
means for a space to be
locally path-connected.
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)
Part (a) places \(M\) among the locally
path-connected spaces, and for those spaces
the path components are exactly the components,
which is (c). The same proposition records that such a space is
connected precisely when it is path-connected, which is (b).
(d)
By (c), the connected components of \(M\) are
the same as the path components, and the proposition cited there also
gives that the components of 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.
The principal content of the proposition is the equivalence of
connectedness and path-connectedness for manifolds. This is 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. That conclusion is 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. 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 \(B \cap B'\), choose one
point, and let \(X\) be the resulting set. The case \(B = B'\) is
included, and its single component is \(B\) itself. 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 \(B\), which 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.
The fundamental groups based at any two points of the same component of
\(M\) are isomorphic, by an isomorphism furnished by conjugation along
a connecting path, as established when the
fundamental group
was introduced. Since \(X\) 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
Lebesgue number lemma
yields a partition
\(0 = a_0 \lt a_1 \lt \cdots \lt a_k = 1\) 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 up to reparametrization as an iterated
product of paths,
\(f \sim f_1 \cdot f_2 \cdots f_k\).
At each interior division point we have
\(f(a_i) \in B_i \cap B_{i+1}\). 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 \(\bar{g}_i\) leaves the path class of \(f\)
unchanged, because the composite \(g_i \cdot \bar{g}_i\) is
path-homotopic
to a constant path by the same retracing that inverts a loop. The
splicing regroups the loop as
\[
f \sim \tilde{f}_1 \cdot \tilde{f}_2 \cdots \tilde{f}_k,
\quad
\tilde{f}_i = g_{i-1} \cdot f_i \cdot \bar{g}_i,
\]
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
\(x_{i-1} \to f(a_{i-1}) \to f(a_i) \to x_i\).
The ball \(B_i\) is homeomorphic to an open ball in \(\mathbb{R}^n\),
which is convex and therefore
star-shaped and simply connected.
Since
a homeomorphism induces an isomorphism of fundamental groups,
the ball \(B_i\) is simply connected as well. Any two paths in
\(B_i\) with the same endpoints are then path-homotopic, because for
such paths \(u\) and \(v\) the loop \(u \cdot \bar{v}\) is
path-homotopic to a constant, and cancelling \(\bar{v}\) against
\(v\) leaves \(u \sim v\). 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.
The countability is a genuine restriction. The fundamental group of a
manifold may well be infinite, as 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. The condition at issue is
local compactness,
which asks every point to have a neighborhood contained in a compact
subset. For Hausdorff spaces 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. The Hausdorff setting is the one that concerns us, since manifolds
are Hausdorff by definition. 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 first 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 \(V\) is a
neighborhood of \(p\) contained in the compact set
\(\overline{V}\), which is the defining condition of local
compactness. Since \(p\) was arbitrary, \(M\) is locally
compact.
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 the consequences it later delivers. The
most important of these is 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
delivers local compactness, and local compactness in turn delivers an
exhaustion of the manifold by compact sets. 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, since 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 \(X\), so 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 \(p \in \bigcup_{X \in \mathcal{X}} \overline{X}\).
Compact Exhaustions
The proof of paracompactness for manifolds proceeds by organizing \(M\)
into an increasing sequence of compact pieces. An
exhaustion by compact sets
is a sequence \((K_i)_{i = 1}^\infty\) of compact subsets with
\(M = \bigcup_{i} K_i\) 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.
Manifolds carry such an exhaustion, and producing one costs nothing beyond
what is already in hand. A manifold is Hausdorff and second-countable by
definition, and locally compact by the proposition above, which are exactly
the hypotheses under which
an exhaustion by compact sets exists.
The construction there builds the \(K_i\) from a basis of precompact open
sets, and on a manifold the basis of precompact coordinate balls is
available for the purpose.
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
\[
V_j = K_{j+1} \setminus \operatorname{int}(K_j), \quad
W_j = \operatorname{int}(K_{j+2}) \setminus K_{j-1}.
\]
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. The set \(K_{j-1}\) 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
\(W_{j'} \cap K_{j'-1} = \emptyset\). A common point therefore 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
\(j + 2 \gt j' - 1\), 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.
Significance
The paracompactness theorem is the technical engine of much of what follows
in manifold theory. Its most important immediate consequence 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. That consequence is developed later in this series, once
smooth maps are in hand.
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.