Question

An open cover \(\mathcal{U}\) of a topological space \(X\) is called locally finite if each \(x \in X\) has a neighborhood that intersects only finitely many members of \(U\). If \(U, \mathcal{V}\) are open covers of \(X, V\) is a refinement of \(U\) if for each \(V \in V\) there exists \(U \in \mathcal{U}\) with \(V \subset U . X\) is called paracompact if every open cover of \(X\) has a locally finite refinement. a. If \(X\) is a \(\sigma\)-compact LCH space, then \(X\) is paracompact. In fact, every open cover \(U\) has locally finite refinements \(\left\\{V_{\alpha}\right\\},\left\\{W_{\alpha}\right\\}\) such that \(\bar{V}_{\alpha}\) is compact and \(\bar{W}_{\alpha} \subset V_{\alpha}\) for all \(\alpha\). (Let \(\left\\{U_{n}\right\\}_{1}^{\infty}\) be as in Proposition 4.39. For each \(n\), \(\left\\{E \cap\left(U_{n+2} \backslash \bar{U}_{n-1}\right): E \subset U\right\\}\) is an open cover of \(\bar{U}_{n+1} \backslash U_{n}\). Choose a finite

Short answer

Yes, \(X\) is paracompact.

Step-by-step solution

Understand the Definitions

First, recall that an open cover \(\mathcal{U}\) of a topological space \(X\) is called locally finite if every point \(x \in X\) has a neighborhood intersecting only finitely many sets from \(\mathcal{U}\). A refinement \(\mathcal{V}\) of an open cover \(\mathcal{U}\) means for each \(V \in \mathcal{V}\), there exists a \(U \in \mathcal{U}\) such that \(V \subset U\). A space \(X\) is paracompact if every open cover has a locally finite refinement.

Define the Context for \(\sigma\)-Compact LCH Space

A \(\sigma\)-compact space is a space that can be expressed as a countable union of compact subspaces. Locally Compact Hausdorff (LCH) spaces are spaces where every point has a compact neighborhood. The problem requires showing that a \(\sigma\)-compact LCH space is paracompact.

Utilize Given Proposition

As per the given hint, use Proposition 4.39, which states that for a \(\sigma\)-compact space \(X\), there exist open sets \(\{U_n\}_{n=1}^{\infty}\) with \(\overline{U}_{n} \subset U_{n+1}\) that serve as a basis for \(X\). Also, note that for each \(n\), \(\{E \cap (U_{n+2} \backslash \overline{U}_{n-1}): E \subset U\}\) forms an open cover of \(\overline{U}_{n+1} \backslash U_{n}\).

Construct Locally Finite Refinements

For each \(n\), use the fact that \(\overline{U}_{n+1} \backslash U_n\) is compact (because it is a closed subset of the compact \(\overline{U}_{n+1}\)) to extract a finite subcover of \(\{E \cap (U_{n+2} \backslash \overline{U}_{n-1}) : E \subset \mathcal{U}\}\). By repeating this process for each \(n\), create a union of all such subcovers, forming a refinement \(\mathcal{V}\) that is locally finite.

Confirm Properties of Refinements

Verify the properties: \(\bar{V}_\alpha\) is compact because it is taken from a finite subcover of a compact set, and \(\bar{W}_\alpha \subset V_\alpha\) follows from the finiteness and proper construction of the refinements overlapping with \(\overline{U}_{n+1} \backslash U_n\).

Key concepts

Locally Finite Refinement

Paracompact spaces have a special type of open cover known as a "locally finite refinement." This concept is crucial in topology as it ensures every point in a space has limited interaction with covers. To understand better, let's break it down.

In any topological space, an open cover \(\mathcal{U}\) is a collection of open sets such that each point in the space is included in at least one of these sets. Now, we're interested in refining this cover. A refinement \(\mathcal{V}\) of \(\mathcal{U}\) means that each set \(V \in \mathcal{V}\) fits into some \(U \in \mathcal{U}\).

• "Locally finite" means around every point \(x \in X\), there exists a neighborhood intersecting only finitely many sets in the cover.
• Essentially, this keeps the cover manageable and avoids overlaps that extend endlessly.

Understanding and constructing locally finite refinements allows mathematicians to work with infinite spaces as though they have finite constraints. This is pivotal in proofs, such as establishing paracompactness where every open cover of a space must have a locally finite refinement.

Sigma-Compact

Sigma-compactness is a delightful concept in topology that deals with organizing spaces into more manageable portions. A topological space is called \(\sigma\)-compact if it can be expressed as a countable union of compact subspaces.

• Compactness, in essence, means that a space can be "covered" by a finite number of small patches.
• In a \(\sigma\)-compact space, rather than a finite number, we have a countable series of such compact spaces coming together.

To visualize, imagine a large quilt made of neatly sewn patches. Each patch represents a compact subspace, and when these patches are arranged, they form the entire quilt, or space. The importance of\(\sigma\)-compactness lies in its usefulness in simplifying complex spaces into an infinite yet countable series of compact ones.

For locally compact Hausdorff spaces, \(\sigma\)-compactness helps ensure that even infinite spaces are still manageable, especially concerning covering properties and refinements.

Locally Compact Hausdorff Space

Locally compact Hausdorff spaces, often abbreviated as LCH spaces, combine specific topological qualities that make them particularly nice to work with.

Firstly, local compactness means that every point in the space has a neighborhood that behaves like a "small" space, i.e., it is compact. This provides a sense of finiteness and controlled behavior around each point.

• Secondly, a Hausdorff space is one where any two distinct points have disjoint neighborhoods, guaranteeing separation and clarity.
• This combination ensures that not only is the space "nicely" separated, but it also allows "compact" handling around each point.

Being both locally compact and Hausdorff facilitates easier manipulation and understanding of spaces. This is particularly advantageous when dealing with paracompactness, as these properties can be exploited to derive locally finite refinements, crucial for demonstrating broader topological characteristics like paracompactness.