description/proof of that 2nd-countable locally compact Hausdorff topological space is \(\sigma\)-compact and paracompact
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of 2nd-countable topological space.
- The reader knows a definition of locally compact topological space.
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of \(\sigma\)-compact topological space.
- The reader knows a definition of paracompact topological space.
- The reader admits the proposition that any locally compact Hausdorff topological space is paracompact if and only if the space is the union of some disjoint open \(\sigma\)-compact subspaces.
- The reader admits the proposition that for any Hausdorff topological space, if and only if each point has a compact neighborhood, the space is locally compact.
- The reader admits the proposition that any compact subset of any Hausdorff topological space is closed.
- The reader admits the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
- The reader admits the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset.
- The reader admits the proposition that any closed subset of any compact topological space is compact.
- The reader admits the proposition that for any topological space, any compact subset of any subspace is compact on the base space.
- The reader admits the proposition that any subset of any countable set is countable.
- The reader admits the proposition that for any infinite set, if there is a surjection from the natural numbers set onto the set, there is a bijection from the natural numbers set onto the set.
Target Context
- The reader will have a description and a proof of the proposition that any 2nd-countable locally compact Hausdorff topological space is \(\sigma\)-compact and paracompact.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(T\): \(\in \{\text{ the 2nd-countable locally compact Hausdorff topological spaces }\}\)
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Statements:
\(T \in \{\text{ the } \sigma \text{ -compact topological spaces }\}\)
\(\land\)
\(T \in \{\text{ the paracompact topological spaces }\}\)
//
2: Proof
Whole Strategy: Step 1: take any countable basis, \(B\), for each \(t \in T\), take a compact neighborhood of \(t\), \(K_t\), and an open neighborhood of \(t\), \(U_t\), from the countable basis, and see that \(\overline{U_t}\) is a compact subset of \(T\); Step 2: apply the proposition that any locally compact Hausdorff topological space is paracompact if and only if the space is the union of some disjoint open \(\sigma\)-compact subspaces.
Step 1:
Let us take any countable basis of \(T\), \(B\).
Let \(t \in T\) be any.
There is a compact neighborhood of \(t\), \(K_t \subseteq T\), by the proposition that for any Hausdorff topological space, if and only if each point has a compact neighborhood, the space is locally compact.
There is an open neighborhood of \(t\), \(U_t \in B\), such that \(U_t \subseteq K_t\), by the definition of basis of topological space.
\(K_t\) is closed on \(T\), by the proposition that any compact subset of any Hausdorff topological space is closed.
\(\overline{U_t} \subseteq K_t\), because the closure is the intersection of all the closed subsets that contain \(U_t\) while \(K_t\) is one of such closed subsets.
\(K_t\) is a compact subspace of \(T\), by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
\(\overline{U_t} \subseteq K_t\) is closed on \(K_t\), by the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset.
\(\overline{U_t}\) is compact on \(K_t\), by the proposition that any closed subset of any compact topological space is compact.
\(\overline{U_t}\) is compact on \(T\), by the proposition that for any topological space, any compact subset of any subspace is compact on the base space.
Let us take the set, \(\{\overline{U_t} \vert t \in T\}\).
Note that as any set does not have any duplication in the elements by the definition of set, \(\{\overline{U_t} \vert t \in T\}\) has no duplication in the elements: although it may seem like to have the number of the points of \(T\) elements, \(\overline{U_t} = \overline{U_{t'}}\) for some \(t, t' \in T\) such that \(t \neq t'\), and \(\overline{U_t} = \overline{U_{t'}}\) is a single element of \(\{\overline{U_t} \vert t \in T\}\).
As each \(\overline{U_t}\) corresponds to \(U_t \in B\), \(\{\overline{U_t} \vert t \in T\}\) is a countable set: \(\{U_t \vert t \in T\} \subseteq B\) is a countable set, by the proposition that any subset of any countable set is countable, and if \(\{U_t \vert t \in T\}\) is finite, \(\{\overline{U_t} \vert t \in T\}\) is finite, and otherwise, there is a bijection, \(f: \mathbb{N} \to \{U_t \vert t \in T\}\), and there is the surjection, \(f': \mathbb{N} \to \{\overline{U_t} \vert t \in T\}, n \mapsto f (n) \mapsto \overline{f (n)}\), and there is a bijection, \(f': \mathbb{N} \to \{\overline{U_t} \vert t \in T\}\), by the proposition that for any infinite set, if there is a surjection from the natural numbers set onto the set, there is a bijection from the natural numbers set onto the set.
\(T = \cup \{\overline{U_t} \vert t \in T\}\), because for each \(t \in T\), \(t \in \overline{U_t}\).
So, \(T\) is \(\sigma\)-compact.
Step 2:
By the proposition that any locally compact Hausdorff topological space is paracompact if and only if the space is the union of some disjoint open \(\sigma\)-compact subspaces, \(T\) is paracompact: any \(\sigma\)-compact topological space is an open \(\sigma\)-compact subspace of itself.
