description/proof of that for topological space and subset of subspace, if subspace is closed, closure of subset on subspace is closure on base space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of closure of subset of topological space.
- The reader knows a definition of subspace topology of subset of topological space.
- The reader knows a definition of closed subset of topological space.
- The reader admits the proposition that for any topological space and any subset of any subspace, the closure of the subset on the subspace is the intersection of the closure of the subset on the base space and the subspace.
- The reader admits the proposition that for any topological space and any subset of any subspace, if the closure of the subset on the subspace is closed on the base space, the closure is the closure of the subset on the base space.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space and any subset of any subspace, if the subspace is closed on the base space, the closure of the subset on the subspace is the closure of the subset on 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: Structured Description
Here is the rules of Structured Description.
Entities:
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Statements:
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2: Proof
Whole Strategy: Step 1: see that
Step 1:
Step 2: