definition of \(T_1\) topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of neighborhood of point on topological space.
Target Context
- The reader will have a definition of \(T_1\) 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:
\(*T\): \(\in \{\text{ the topological spaces }\}\)
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Conditions:
\(\forall \{t_1, t_2\} \subseteq T \text{ such that } t_1 \neq t_2 (\exists U_{t_1} \in \{\text{ the open neighborhoods of } t_1\} (t_2 \notin U_{t_1}))\)
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As it holds also for \(\{t_2, t_1\} \subseteq T\), there is inevitably also \(U_{t_2}\) such that \(t_1 \notin U_{t_2}\): this is not about that there is \(U_{t_1}\) or \(U_{t_2}\).
2: Note
This definition equals that each point is a closed subset of \(T\).
Let us see that fact.
Let us suppose this definition.
Let \(t_1 \in T\) be any.
Let \(t_2 \in T \setminus \{t_1\}\) be any.
\(t_1 \neq t_2\).
So, there is an open neighborhood of \(t_2\), \(U_{t_2} \subseteq T\), such that \(t_1 \notin U_{t_2}\), which means that \(U_{t_2} \subseteq T \setminus \{t_1\}\).
So, \(T \setminus \{t_1\} \subseteq T\) is an open subset, by the local criterion for openness.
So, \(\{t_1\}\) is a closed subset of \(T\).
Let us suppose that each point is a closed subset of \(T\).
Let \(\{t_1, t_2\} \subseteq T\) be any such that \(t_1 \neq t_2\).
\(t_1 \in T \setminus \{t_2\}\).
\(T \setminus \{t_2\}\) is open, so, there is an open neighborhood of \(t_1\), \(U_{t_1} \subseteq T\), such that \(U_{t_1} \subseteq T \setminus \{t_2\}\), by the local criterion for openness.
That means that \(t_2 \notin U_{t_1}\).
