description/proof of that for topological space and subset, interior of subset is set of points that have open neighborhoods contained in subset
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of interior of subset of topological space.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space and any subset, the interior of the subset is the set of the points of the space that have some open neighborhoods contained in the subset.
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:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(S\): \(\subseteq T\)
\(Int (S)\): \(= \text{ the interior of } S \text{ on } T\)
\(S^`\): \(= \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq S)\}\)
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Statements:
\(Int (S) = S^`\)
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2: Proof
Whole Strategy: Step 1: see that \(Int (S) \subseteq S^`\); Step 2: see that \(S^` \subseteq Int (S)\); Step 3: conclude the proposition.
Step 1:
Let \(t \in Int (S)\) be any.
As \(Int (S)\) is the union of the open subsets contained in \(S\), \(t \in U\) for an open subset, \(U \subseteq T\), such that \(U \subseteq S\).
So, \(U_t := U \subseteq T\) is an open neighborhood of \(t\) such that \(U_t \subseteq S\).
So, \(t \in S^`\).
So, \(Int (S) \subseteq S^`\).
Step 2:
Let \(t \in S^`\) be any.
There is a \(U_t\) such that \(U_t \subseteq S\).
As \(Int (S)\) is the union of the open subsets contained in \(S\), \(U_t \subseteq Int (S)\).
\(t \in U_t \subseteq Int (S)\).
So, \(S^` \subseteq Int (S)\).
Step 3:
So, \(Int (S) = S^`\).
