definition of basis of topological space
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of neighborhood of point.
Target Context
- The reader will have a definition of basis of 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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\(*B\): \(\subseteq Pow (T)\)
//
Conditions:
\(\forall U \in B (U \in \{\text{ the open subsets of } T\})\)
\(\land\)
\(\forall t \in T, \forall N_t \subseteq T \in \{\text{ the neighborhoods of } t\} (\exists U \in B (t \in U \subseteq N_t))\)
//
2: Natural Language Description
For any topological space, \(T\), any set of some open sets such that for each neighborhood, \(N_t\), of each point, \(t \in T\), there is an open set, \(U \in B\), such that \(t \in U \subseteq N_t\)
3: Note
When the topology of \(T\) is not determined yet, any basis determines the topology, by the proposition that any basis of any topological space determines the topology.
