Chapter 16
Stone’s TheoremThis section is devoted to Stone’s theorem which says that a metric space is paracompact,defined below. See [98] for this which is where I read it. First is the definition of what ismeant by a refinement.
Definition 16.0.1 Let S be a topological space. We say that a collection of sets D is arefinement of an open cover S, if every set of D is contained in some set of S. An openrefinement would be one in which all sets are open, with a similar convention holding forthe term “ closed refinement”.
Definition 16.0.2 We say that a collection of sets D, is locally finite if for all p ∈ S, thereexists V an open set containing p such that V has nonempty intersection with only finitelymany sets of D.
Definition 16.0.3 We say S is paracompact if it is Hausdorff and for every open cover S,there exists an open refinement D such that D is locally finite and D covers S.
Theorem 16.0.4 If D is locally finite then
∪{D : D ∈D}= ∪{D : D ∈D}.
Proof: It is clear the left side is a subset of the right. Let p be a limit point of
∪{D : D ∈D}
and let p ∈V , an open set intersecting only finitely many sets of D, D1...Dn. If p is not inany of Di then p ∈W where W is some open set which contains no points of ∪n
i=1Di. ThenV ∩W contains no points of any set of D and this contradicts the assumption that p is alimit point of
∪{D : D ∈D}.Thus p ∈ Di for some i.
We say S⊆P (S) is countably locally finite if
S= ∪∞n=1Sn
and each Sn is locally finite. The following theorem appeared in the 1950’s. It will be usedto prove Stone’s theorem.
Theorem 16.0.5 Let S be a regular topological space. (If p ∈U open, then there exists anopen set V such that p ∈ VĚ„ ⊆U. ) The following are equivalent
1.) Every open covering of S has a refinement that is open, covers S and is countablylocally finite.
2.) Every open covering of S has a refinement that is locally finite and covers S. (Thesets in refinement maybe not open.)
3.) Every open covering of S has a refinement that is closed, locally finite, and coversS. (Sets in refinement are closed.)
4.) Every open covering of S has a refinement that is open, locally finite, and covers S.(Sets in refinement are open.)
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