description/proof of that normal topological space is regular
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of normal topological space.
- The reader knows a definition of regular topological space.
Target Context
- The reader will have a description and a proof of the proposition that any normal topological space is regular.
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:
\(T\): \(\in \{\text{ the normal topological spaces }\}\)
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Statements:
\(T \in \{\text{ the regular topological spaces }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(T\) satisfies the conditions to be regular.
Step 1:
\(\forall t \in T (\{t\} \in \{\text{ the closed subsets of } T\})\) is directly required by the definition of normal topological space.
Let \(t \in T\) be any.
Let \(C \subseteq T\) be any closed subset such that \(t \notin C\).
\(\{t\} \subseteq T\) is closed.
\(\{t\} \cap C = \emptyset\).
There are an open neighborhood of \(\{t\}\), \(U_{\{t\}} \subseteq T\), and an open neighborhood of \(C\), \(U_C \subseteq T\), such that \(U_{\{t\}} \cap U_C = \emptyset\).
But \(U_{\{t\}}\) is an open neighborhood of \(t\).
So, \(T\) is regular.
