A description/proof of that basis determines topology
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 basis of topological space.
- The reader admits some criteria for any collection of open sets to be a basis.
Target Context
- The reader will have a description and a proof of the proposition that any basis of any topological space determines the topology.
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: Description
Any basis, \(\{B_\alpha\}\), of any topological space, \(T\), determines the topology of \(T\).
2: Proof
For any open set, \(U\), on \(T\), around any point, \(p \in U\), there is an open set, \(U_p \subseteq U\), by the local criterion for openness. There is an open set, \(B_\alpha \subseteq U_p\), in the basis, by the definition of basis. \(U\) is the union of such open sets in the basis, by some criteria for any collection of open sets to be a basis.
Supposing that there are 2 topologies of the same set as the topological spaces, \(T\) and \(T'\), with the same basis, any open set, \(U\), on \(T\) is the union of some open sets in the basis, so, \(U\) is also an open set on \(T'\), and likewise, any open set, \(U'\), on \(T'\) is also an open set on \(T\), so, \(T = T'\), which is what this proposition means.
3: Note
So, the topological can be said to 'be generated by the basis'.
