definition of normal 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 subset of topological space.
- The reader knows a definition of neighborhood of subset of topological space.
Target Context
- The reader will have a definition of normal 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:
\(*T\): \(\in \{\text{ the topological spaces }\}\)
//
Conditions:
\(\forall t \in T (\{t\} \in \{\text{ the closed subsets of } T\})\)
\(\land\)
\(\forall C_1, C_2 \in \{\text{ the closed subsets of } T\} \text{ such that } C_1 \cap C_2 = \emptyset (\exists U_{C_1} \in \{\text{ the open neighborhoods of } C_1\}, \exists U_{C_2} \in \{\text{ the open neighborhoods of } C_2\} (U_{C_1} \cap U_{C_2} = \emptyset))\)
2: Note
Some people define 'normal topological space' without the requirement, "\(\forall t \in T (\{t\} \in \{\text{ the closed subsets of } T\})\)".
