CW implies normal
This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., CW-space) must also satisfy the second topological space property (i.e., normal space)
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Every CW-space (viz every space which can be given a CW-complex structure) is normal, viz it is Hausdorff and any two disjoint closed sets can be separated by disjoint open sets. (note that Hausdorffness follows from one of the definitions, so we only prove normality).
Goal of the proof
Let be a CW-space. Equip with a CW-complex structure and let denote the -skeleton with respect to that structure.
Let be closed subsets. The goal is to construct a function such that and (this will show that and are separated by disjoint open sets).
To construct this , we construct a family of functions on the -skeletons, such that restricted to is for , and such that and .
Suppose we have constructed which takes the value on and on . We need to use this to define on , which extends .
Now we extend the function separately on the interior of each disc. Note that the choice of how we extend the function on the interior of one disc, does not affect the choice on the interior of any other disc.
For a disc corresponding to a cell attached via a map , we have the following data:
- A map
- A subset of , which is , along with all points on whose image via lies inside
- A corresponding subset of
Consider as a subset of . This is a closed subset. Define such that , and . The well-definedness and continuity of follow from the properties of , the fact that all three subsets are closed, and the gluing lemma.
Since is normal, the function extends to a continuous map . The restriction of to the interior of , is the extension we require.