definition of neighborhoods basis at point on 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 neighborhood of point on topological space.
Target Context
- The reader will have a definition of neighborhoods basis at point on 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:
\( T\): \(\in \{\text{ the topological spaces }\}\), with any topology, \(O\)
\( t\): \(\in T\)
\(*B_t\): \(\subseteq \{\text{ the neighborhoods of } t \text{ on } T\}\)
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Conditions:
\(\forall N'_t \in \{\text{ the neighborhoods of } t \text{ on } T\} (\exists N_t \in B_t (N_t \subseteq N'_t))\)
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2: Note
\(B_t\) does not need to be a set of open neighborhoods of \(t\) although it may be a set of open neighborhoods of \(t\).
When \(B_t\) is a set of open neighborhoods of \(t\), it is called "open neighborhoods basis at \(t\) on \(T\)".
