A definition of regular topological space
Topics
About: topological space
The table of contents of this article
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.
Target Context
- The reader will have a definition of regular 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: 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 \(C \subseteq N_2 \subseteq T\), \(N_1 \cap N_2 = \emptyset\)
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.
