definition of closure 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 closed set.
- The reader knows a definition of intersection of set.
Target Context
- The reader will have a definition of closure 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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( S\): \(\subseteq T\)
\(\overline{S}\) = \(\cap \{C \in \{\text{ the closed subsets of } T\} \vert S \subseteq C\}\)
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Conditions:
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2: Note
\(\overline{S}\) is a closed subset of \(T\) containing \(S\): it is closed as an intersection of closed subsets of \(T\); \(S \subseteq \overline{S}\), because for each \(s \in S\), \(s \in C\) for each \(C\), and \(s \in \overline{S}\).
Colloquially, it is called "the smallest closed subset containing \(S\)", which is indeed warranted, because while \(\overline{S}\) is a closed subset containing \(S\), for any closed subset containing \(S\), \(C \subseteq T\) such that \(S \subseteq C\), \(\overline{S} \subseteq C\), because \(\overline{S}\) is the intersection of all the such closed subsets.
