definition of refinement of open cover of 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 refinement of open cover of 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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( S\): \(\subseteq T\)
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{U_j \in \{\text{ the open subsets of } T\} \vert j \in J\}\): such that \(S \subseteq \cup_{j \in J} U_j\)
\( L\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(*\{V_l \in \{\text{ the open subsets of } T\} \vert l \in L\}\): such that \(S \subseteq \cup_{l \in L} V_l\)
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Conditions:
\(\forall l \in L (\exists j \in J (V_l \subseteq U_j))\)
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2: Note
Some people may require \(\{V_l \vert l \in L\}\) to be just a cover of \(S\) instead of being an open cover by "refinement" and then, they will say "open refinement" for this definition.
This definition requires \(\{V_l \vert l \in L\}\) to be an open cover, because we usually naturally expect an open cover for any open cover, \(\{U_j \vert j \in J\}\).
