A description/proof of that topological subspace is locally closed iff it is intersection of closed subset and open subset of base space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of locally closed topological space.
- The reader knows a definition of subspace topology.
- The reader admits the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset.
- The reader admits the proposition that for any topological space, the intersection of the closure of any subset and any open set is contained in the closure of the intersection of the subset and the open set.
Target Context
- The reader will have a description and a proof of the proposition that any topological subspace is locally closed if and only if it is the intersection of a closed subset and an open subset of the base 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: Description
For any topological space,
2: Proof
Let us suppose that
Let us suppose that