Paracompact spaces are sometimes required to also be
Hausdorff to extend their properties. • (
Theorem of Jean Dieudonné) Every paracompact Hausdorff space is
normal. • Every paracompact Hausdorff space is a
shrinking space, that is, every open cover of a paracompact Hausdorff space has a shrinking: another open cover indexed by the same set such that the closure of every set in the new cover lies inside the corresponding set in the old cover. • On paracompact Hausdorff spaces,
sheaf cohomology and
Čech cohomology are equal.
Partitions of unity The most important feature of paracompact
Hausdorff spaces is that they admit
partitions of unity subordinate to any open cover. This means the following: if
X is a paracompact Hausdorff space with a given open cover, then there exists a collection of
continuous functions on
X with values in the
unit interval [0, 1] such that: • for every function
f:
X →
R from the collection, there is an open set
U from the cover such that the
support of
f is contained in
U; • for every point
x in
X, there is a neighborhood
V of
x such that all but finitely many of the functions in the collection are identically 0 in
V and the sum of the nonzero functions is identically 1 in
V. In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any open cover (see
below). This property is sometimes used to define paracompact spaces (at least in the Hausdorff case). Partitions of unity are useful because they often allow one to extend local constructions to the whole space. For instance, the integral of
differential forms on paracompact
manifolds is first defined locally (where the manifold looks like
Euclidean space and the integral is well known), and this definition is then extended to the whole space via a partition of unity.
Proof that paracompact Hausdorff spaces admit partitions of unity {{hidden :
Lemma 1: If \mathcal{O}\, is a locally finite open cover, then there exists open sets W_{U}\, for each U\in\mathcal{O}\,, such that each \bar{W_{U}}\subseteq U\, and \{W_{U}:U\in\mathcal{O}\}\, is a locally finite refinement. :
Lemma 2: If \mathcal{O}\, is a locally finite open cover, then there are continuous functions f_{U}:X\to[0,1]\, such that \operatorname{supp}~f_{U}\subseteq U\, and such that f:=\sum_{U\in\mathcal{O}}f_{U}\, is a continuous function which is always non-zero and finite. :
Theorem: In a paracompact Hausdorff space X\,, if \mathcal{O}\, is an open cover, then there exists a partition of unity subordinate to it. :
Proof (Lemma 1): :Let \mathcal{V}\, be the collection of open sets meeting only finitely many sets in \mathcal{O}\,, and whose closure is contained in a set in \mathcal{O}. One can check as an exercise that this provides an open refinement, since paracompact Hausdorff spaces are regular, and since \mathcal{O}\, is locally finite. Now replace \mathcal{V}\, by a locally finite open refinement. One can easily check that each set in this refinement has the same property as that which characterised the original cover. : Now we define W_{U}=\bigcup\{A\in\mathcal{V}:\bar{A}\subseteq U\}\,. The property of \mathcal{V}\, guarantees that every A\in\mathcal{V} is contained in some W_U. Therefore \{W_{U}:U\in\mathcal{O}\}\, is an open refinement of \mathcal{O}\,. Since we have W_{U} \subseteq U , this cover is immediately locally finite. : : Now we want to show that each \bar{W_{U}}\subseteq U\,. For every x \notin U, we will prove that x \notin \bar{W_U} . Since we chose \mathcal{V} to be locally finite, there is a neighbourhood V[x] of x such that only finitely many sets in \mathcal{V} have non-empty intersection with V[x] , and we note A_{1},...,A_{n},... \in \mathcal{V} those in the definition of W_U . Therefore we can decompose W_U in two parts: A_{1},...,A_{n} \in \mathcal{V} who intersect V[x] , and the rest A \in \mathcal{V} who don't, which means that they are contained in the closed set C:= X\setminus V[x] . We now have \bar{W_U} \subseteq \bar{A_1} \cup...\cup \bar{A_n}\cup C . Since \bar{A_i} \subseteq U and x \notin U , we have x \notin \bar{A_i} for every i . And since C is the complement of a neighbourhood of x , x is also not in C . Therefore we have x \notin \bar{W_U} . :
Proof (Lemma 2): :Applying Lemma 1, let f_{U}:X\to[0,1]\, be continuous maps with f_{U}\upharpoonright\bar{W}_{U}=1\, and \operatorname{supp}~f_{U}\subseteq U\, (by Urysohn's lemma for disjoint closed sets in normal spaces, which a paracompact Hausdorff space is). Note by the support of a function, we here mean the points not mapping to zero (and not the closure of this set). To show that f=\sum_{U\in\mathcal{O}}f_{U}\, is always finite and non-zero, take x\in X\,, and let N\, a neighbourhood of x\, meeting only finitely many sets in \mathcal{O}\,; thus x\, belongs to only finitely many sets in \mathcal{O}\,; thus f_{U}(x)=0\, for all but finitely many U\,; moreover x\in W_{U}\, for some U\,, thus f_{U}(x)=1\,; so f(x)\, is finite and \geq 1\,. To establish continuity, take x,N\, as before, and let S=\{U\in\mathcal{O}:N\text{ meets }U\}\,, which is finite; then f\upharpoonright N=\sum_{U\in S}f_{U}\upharpoonright N\,, which is a continuous function; hence the preimage under f\, of a neighbourhood of f(x)\, will be a neighbourhood of x\,. :
Proof (Theorem): :Take \mathcal{O}^*\, a locally finite subcover of the refinement cover: \{V\text{ open }:(\exists{U\in\mathcal{O}})\bar{V}\subseteq U\}\,. Applying Lemma 2, we obtain continuous functions f_{W}:X\to[0,1]\, with \operatorname{supp}~f_{W}\subseteq W\, (thus the usual closed version of the support is contained in some U\in\mathcal{O}\,, for each W\in\mathcal{O}^*\,; for which their sum constitutes a
continuous function which is always finite non-zero (hence 1/f\, is continuous positive, finite-valued). So replacing each f_{W}\, by f_{W}/f\,, we have now — all things remaining the same — that their sum is everywhere 1\,. Finally for x\in X\,, letting N\, be a neighbourhood of x\, meeting only finitely many sets in \mathcal{O}^*\,, we have f_{W}\upharpoonright N=0\, for all but finitely many W\in\mathcal{O}^*\, since each \operatorname{supp}~f_{W}\subseteq W\,. Thus we have a partition of unity subordinate to the original open cover. }} == Relationship with compactness ==