2022-07-31

326: Basis Determines Topology

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A description/proof of that basis determines topology

Topics


About: topological space

The table of contents of this article


Starting Context



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'.


References


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