description/proof of that for topological space, path-connected component is contained in connected component
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological connected component.
- The reader knows a definition of topological path-connected component.
- The reader admits the proposition that any path-connected topological component is exactly any path-connected topological subspace that cannot be made larger.
- The reader admits the proposition that any path-connected topological space is connected.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space, any path-connected component is contained in the corresponding connected component.
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:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(C\): \(\in \{\text{ the path-connected components of } T\}\)
\(C'\): \(\in \{\text{ the connected components of } T\}\), such that \(c \in C'\) for any fixed \(c \in C\)
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Statements:
\(C \subseteq C'\)
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2: Note
\(C'\) is uniquely determined, by this proposition, because for any other \(\widetilde{C'}\) taken based on \(\widetilde{c} \in C\), \(C \subseteq \widetilde{C'}\) means that \(c \in \widetilde{C'}\), so, \(\widetilde{C'}\) is the connected component such that \(c \in \widetilde{C'}\), which is \(C'\).
3: Proof
Whole Strategy: Step 1: see that \(C\) is a path-connected subspace; Step 2: see that \(C\) is a connected subspace; Step 3: see that \(C \subseteq C'\).
Step 1:
\(C\) is a path-connected topological subspace, by the proposition that any path-connected topological component is exactly any path-connected topological subspace that cannot be made larger.
Step 2:
\(C\) is a connected topological subspace, by the proposition that any path-connected topological space is connected.
Step 3:
So, for each \(c' \in C\), \(c\) and \(c'\) are connected, because they are contained in the connected subspace, \(C\), which means that \(C \subseteq C'\).
