description/proof of that regular topological space is Hausdorff
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of regular topological space.
- The reader knows a definition of Hausdorff topological space.
Target Context
- The reader will have a description and a proof of the proposition that any regular topological space is Hausdorff.
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:
\(T\): \(\in \{\text{ the regular topological spaces }\}\)
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Statements:
\(T \in \{\text{ the Hausdorff topological spaces }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(T\) satisfies the conditions to be Hausdorff.
Step 1:
Let \(t_1, t_2 \in T\) be any such that \(t_1 \neq t_2\).
\(\{t_2\} \subseteq T\) is a closed subset.
\(t_1 \notin \{t_2\}\).
So, there are an open neighborhood of \(t_1\), \(U_{t_1} \subseteq T\), and an open neighborhood of \(\{t_2\}\), \(U_{\{t_2\}} \subseteq T\), such that \(U_{t_1} \cap U_{\{t_2\}} = \emptyset\).
But \(U_{\{t_2\}}\) is an open neighborhood of \(t_2\).
So, \(T\) is Hausdorff.
