definition of compact topological space
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of topological space.
Target Context
- The reader will have a definition of compact 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:
\(*T\): \(\in \{\text{ the topological spaces }\}\)
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Conditions:
\(\forall \{U_\alpha \vert \alpha \in A\}\), where \(A \in \{\text{ the possibly uncountable index sets }\}\), \(U_\alpha \in \{\text{ the open subsets of } T\}\), and \(\cup_{\alpha \in A} U_\alpha = T\)
(
\(\exists J \subseteq A\)
(
\(J \in \{\text{ the finite sets }\}\)
\(\land\)
\(\cup_{j \in J} U_j = T\)
)
)
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2: Natural Language Description
Any topological space, \(T\), such that each open cover of \(T\) has a finite subcover
