definition of interior 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.
- The reader knows a definition of union of set.
Target Context
- The reader will have a definition of interior 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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( S\): \(\subseteq T\)
\(*Int (S)\): \(= \cup \{U \in \{\text{ the open subsets of } T\} \vert U \subseteq S\}\)
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Conditions:
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2: Note
\(Int (S)\) is an open subset of \(T\) contained in \(S\): it is open as a union of open subsets of \(T\); \(Int (S) \subseteq S\), because for each \(p \in Int (S)\), \(p \in U\) for a \(U\), and \(p \in U \subseteq S\).
Colloquially, it is called "the largest open subset contained in \(S\)", which is indeed warranted, because while \(Int (S)\) is an open subset contained in \(S\), for any open subset contained in \(S\), \(U \subseteq T\) such that \(U \subseteq S\), \(U \subseteq Int (S)\), because \(Int (S)\) is the union of all the such open subsets.
