description/proof of that for disjoint union topological space, closure of disjoint union of subsets is disjoint union of closures of subsets
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of disjoint union topology.
- The reader knows a definition of closure of subset of topological space.
- The reader admits the proposition that the closure of any subset is the union of the subset and the accumulation points set of the subset.
Target Context
- The reader will have a description and a proof of the proposition that for any disjoint union topological space, the closure of the disjoint union of any subsets is the disjoint union of the closures of the subsets.
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:
So,
Step 2:
Let
Let
So,
So,
So,
So,
So,