description/proof of that for topological space, intersection of compact subset of space and subspace is not necessarily compact on subspace
Topics
About: topological space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for a topological space, the intersection of a compact subset of the space and a subspace is not necessarily compact on the subspace.
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:
\(T'\): \(\in \{\text{ the topological spaces }\}\)
\(T\): \(\in \{\text{ the topological subspaces of } T'\}\)
\(K\): \(\in \{\text{ the compact subsets of } T'\}\), such that \(\lnot K \subseteq T\)
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Statements:
Not necessarily, \(K \cap T \in \{\text{ the compact subsets of } T\}\)
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2: Note
When \(K\) is contained in \(T\), \(K \cap T = K\) is inevitably compact on \(T\), as is described in an article.
3: Proof
Whole Strategy: Step 1: see a counterexample.
Step 1:
Let us see a counterexample.
Let \(T' = \mathbb{R}\) with the Euclidean topology, \(T = (0, 1)\), and \(K = [-1, 1]\).
\(K\) is compact on \(\mathbb{R}\), by the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded.
\(K \cap T = (0, 1)\) is not compact on \(T\), because the open cover, \(\{(1 / 2, 1), (1 / 4, 1 / 2 + 1 / 4), (1 / 8, 1 / 2 + 1 / 4 + 1 /8), ...\}\), has no finite subcover, for example.
