definition of countably compact subset of topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
Target Context
- The reader will have a definition of countably compact subset of topological 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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\(*S\): \(\subseteq T\)
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Conditions:
\(\forall \{U_{j'} \in \{\text{ the open subsets of } T\} \vert j' \in J'\}\), where \(J' \in \{\text{ the countable index sets }\}\) and \(S \subseteq \cup_{j' \in J'} U_{j'} (\exists J \in \{\text{ the finite subsets of } J'\} (S \subseteq \cup_{j \in J} U_j))\)
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2: Note
\(\{U_{j'} \in \{\text{ the open subsets of } T\} \vert j' \in J'\}\) is called "countable open cover of \(S\)".
Compare with compact subset of topological space.
