August 19, 2010

A metric space is paracompact

Posted by Akhil Mathew under topology | Tags: , |
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Today, I shall use the theorem of Michael discussed earlier to prove that a metric space is paracompact.

Theorem 1 (Stone) A metric space is paracompact.

This theorem seems to use the axiom of choice, or some version thereof, in all proofs.

1. Proof of Stone’s theorem

Suppose given a cover {\mathfrak{A}=\left\{U_\alpha\right\}} of the metric space {X} (with metric {d}, say). We will show that there is a refinement of {\mathfrak{A}} that can be decomposed into a countable collection of locally finite families. Thanks to Michael’s theorem, this will prove the result.
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August 19, 2010

A theorem of Michael on paracompactness

Posted by Akhil Mathew under topology | Tags: |
[5] Comments 

Today I want to discuss an equivalent and seemingly weaker condition of paracompactness due to Ernest Michael (1953, Proc. of the AMS).

Theorem 1 (Michael) Suppose {X} is regular and every open cover {\mathfrak{A}} of {X} has a refinement that can be decomposed into a countable collection of locally finite {\mathfrak{A}_i} collections of open sets. Then {X} is paracompact.

Note that the {\mathfrak{A}_i}‘s are not required to be covers, only locally finite! This is a significant strengthening of the usual definition of paracompactness.

Following Michael’s original paper, I shall discuss the proof of this result. First, I shall give an auxiliary result, of independent result, that yields yet another variation on the theme of paracompactness: we don’t have to require the locally finite refinements to be open.

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August 17, 2010

Paracompactness

Posted by Akhil Mathew under topology | Tags: , |
[5] Comments 

I’ve been reviewing some basic general topology as of late. I will post some of this material here. Apologies to readers who prefer more advanced topics; my current focus is on foundational material.

Often, we’d like to prove that a given subset {A \subset X} of a topological space {X} has some given property, e.g. that it is open or closed. In many cases, however, the big space {X} may not be easily understandable, but local pieces of it may be. For instance, {X} might be a manifold, and we might not know what the global structure of {X} is, but we do know that {X} is locally homeomorphic (or diffeomorphic) to a ball in {\mathbb{R}^n}. So we need a way to go from local results to global results.

Proposition 1 Let {\left\{U_\alpha\right\}} be an open cover of the topological space {X}. Suppose {W \subset X} and {W \cap U_{\alpha}} is open in {U_{\alpha}} for each {\alpha}. Then {W} is open in {X}.

So openness is a local property.

This is the easy result. Indeed, since {U_{\alpha}} is an open set, each {W \cap U_{\alpha} } is open in {X}. But

\displaystyle W = \bigcup W \cap U_{\alpha}

so that {W} is a union of open sets, hence open.

Similarly, we can deduce the corresponding result for closed sets:

Corollary 2 Suppose {U_{\alpha}} is an open cover of {X}. Let {W \subset X}. Suppose {W \cap U_{\alpha}} is closed in { U_{\alpha}} for each {U_{\alpha}}. Then {W} is closed in {X}.

Indeed, this follows from the previous result, with {W} replaced with {X - W}.

However, the analogous result is no longer true if we look at closed covers. Consider for instance the closed cover of {\mathbb{R}^2} by vertical lines. The set {W}, defined as the intersection of the graph {y=x} with the upper right quadrant, is not closed, though its intersection with each vertical line is closed (in fact, is a point). So we need something more. The problem, as we will see, is that there are too many lines. (more…)