367: Regular Topological Space

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A definition of regular topological space

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of closed set.
• The reader knows a definition of neighborhood of point.
• The reader knows a definition of neighborhood of subset.

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Target Context

• The reader will have a definition of regular topological space.

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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.

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Main Body

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1: Definition

Any topological space, $T$, such that each 1 point subset is closed, and for any point, $p\in T$, and any closed set that does not contain $p$, $C\subseteq T,p\notin C$, there are disjoint neighborhoods, $p\in N_1\subseteq T$ and $C\subseteq N_2\subseteq T$, $N_1\cap N_2=\mathrm{\varnothing }$

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2: Note

In fact, $N_1$ and $N_2$ can be taken as open neighborhoods, because we can take open neighborhoods contained in $N_1$ and $N_2$, which preserves the disjointness.

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References

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