2023-06-18

305: Topological Subspace Is Locally Closed iff It Is Intersection of Closed Subset and Open Subset of Base Space

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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



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, T, any subspace, T1T, is locally closed if and only if T1=CU where C and U are a closed subset and an open set, respectively, of T.


2: Proof


Let us suppose that T1=CU. T1=T1U, because obviously T1UCU, and for any point, pCU, pT1 and pU, so, pT1U, so, CPT1U. If UT1 is closed on U, U can be taken to be an open neighborhood of each point, pT1, required for T1 to be locally closed. But UT1 is in fact closed on U, because UT1=UT1U=T1U.

Let us suppose that T1 is locally closed. Let us define U:=pUp where UpT is an open neighborhood of p on T required for T1 to be locally closed, so, UpT1 is closed on Up. U is open on T. Let us define C=T1, closed on T. Then, CU=T1pUp=p(T1Up)=p(T1Up)=T1, where the penultimate equal is because on the one hand, (T1Up)(T1Up)T1Up, by 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, and on the other hand, T1Up=T1UpUp, because as T1Up is closed on Up, there is a closed subset, CpT, such that T1Up=CpUp, but T1UpCp and as Cp is closed, T1UpCp, and in fact, Cp can be taken to be T1Up, so, T1Up=CpUp=T1UpUp, then T1UpUp=T1Up(T1Up)UpT1UpUp while the 1st term and the 3rd term equal, so, the 2nd term equals the 1st term, and the last equal is because U covers T1.


References


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