description/proof of that for Hausdorff topological space, compact subset, and subset contained in compact subset, closure of subset on space is compact on compact subspace and on space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of compact subset of topological space.
- The reader knows a definition of closure of subset of topological space.
- The reader admits the proposition that any compact subset of any Hausdorff topological space is closed.
- 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 the compactness of any topological subset as a subset equals the compactness as a subspace.
- 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.
Target Context
- The reader will have a description and a proof of the proposition that for any Hausdorff topological space, any compact subset, and any subset contained in the compact subset, the closure of the subset on the space is compact on the compact subspace and on the space.
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:
\(T\): \(\in \{\text{ the Hausdorff topological spaces }\}\)
\(K\): \(\in \{\text{ the compact subsets of } T\}\)
\(S\): \(\subseteq T\), such that \(S \subseteq K\)
\(\overline{S}\): \(= \text{ the closure of } S \text{ on } T\)
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Statements:
\(\overline{S} \in \{\text{ the compact subsets of } K\} \cap \{\text{ the compact subsets of } T\}\)
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2: Proof
Whole Strategy: Step 1: see that \(K\) is closed on \(T\); Step 2: see that \(\overline{S} \subseteq K\); Step 3: see that \(\overline{S}\) is closed on \(K\); Step 4: see that \(\overline{S}\) is compact on \(K\); Step 5: see that \(\overline{S}\) is compact on \(T\).
Step 1:
\(K\) is closed on \(T\), by the proposition that any compact subset of any Hausdorff topological space is closed.
Step 2:
\(\overline{S} \subseteq K\), because \(\overline{S}\) is the intersection of the closed subsets that contain \(S\) and \(K\) is one of such the closed subsets.
Step 3:
\(\overline{S}\) is closed on \(T\), and is closed on \(K\), because \(\overline{S} = \overline{S} \cap K\), 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.
Step 4:
\(K\) is a compact subspace, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
\(\overline{S}\) is compact on \(K\), by the proposition that any closed subset of any compact topological space is compact.
Step 5:
\(\overline{S}\) is compact on \(T\), by the proposition that for any topological space, any compact subset of any subspace is compact on the base space.
