description/proof of that quasi-connected component is open on locally connected topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of locally connected topological space.
- The reader knows a definition of topological quasi-connected component.
- The reader admits the proposition that for any topological space, each connected component is contained in the corresponding quasi-connected component.
- The reader admits the proposition that for any locally connected topological space, each connected component is open.
Target Context
- The reader will have a description and a proof of the proposition that any quasi-connected component is open on any locally connected topological 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:
\(T\): \(\in \{\text{ the locally connected topological spaces }\}\)
\(C'\): \(\in \{\text{ the quasi-connected components of } T\}\)
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Statements:
\(C' \in \{\text{ the open subsets of } T\}\)
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2: Proof
Whole Strategy: Step 1: see that \(C' = \cup_{c' \in C'} C_{c'}\) where \(C_{c'}\) is the connected component that contains \(c'\); Step 2: see that \(C_{c'}\) is open; Step 3: conclude the proposition.
Step 1:
\(C' = \cup_{c' \in C'} C_{c'}\) where \(C_{c'}\) is the connected component that contains \(c'\), because \(C_{c'} \subseteq C'\), by the proposition that for any topological space, each connected component is contained in the corresponding quasi-connected component.
Step 2:
Each \(C_{c'} \subseteq T\) is open, by the proposition that for any locally connected topological space, each connected component is open.
Step 3:
So, \(C' = \cup_{c' \in C'} C_{c'}\) is open on \(T\) as the union of some open subsets.
