description/proof of that for topological space and set of connected subspaces, subspace is contained in connected component of union of subspaces
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of connected topological space.
- The reader knows a definition of connected topological component.
- The reader knows a definition of topological subspace.
- The reader admits the proposition that in any nest of topological subspaces, the connected-ness of any subspace does not depend on the superspace of which the subspace is regarded to be a subspace.
- The reader admits the proposition that for any topological space, the union of any connected subspaces that share any point is connected.
- The reader admits the proposition that any connected topological component is exactly any connected topological subspace that cannot be made larger.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space and any set of connected subspaces, each subspace is contained in a connected component of the union of the subspaces.
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:
\(T'\): \(\in \{\text{ the topological spaces }\}\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{T_j \vert j \in J\}\): \(T_j \in \{\text{ the connected topological subspaces of } T'\}\)
\(T\): \(= \cup_{j \in J} T_j \subseteq T'\) with the subspace topology
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Statements:
\(\forall j \in J (\exists C \in \{\text{ the connected components of } T\} (T_j \subseteq C))\)
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2: Proof
Whole Strategy: Step 1: see that \(T_j\) is a connected subspace of \(T\); Step 2: conclude the proposition.
Step 1:
\(T_j\) is a connected subspace of \(T\), by the proposition that in any nest of topological subspaces, the connected-ness of any subspace does not depend on the superspace of which the subspace is regarded to be a subspace.
Step 2:
Let us take the union of all the connected subspaces of \(T\) that contain \(T_j\), which is a connected subspace of \(T\) that cannot be made larger, by the proposition that for any topological space, the union of any connected subspaces that share any point is connected: that cannot be made larger because if there was a larger one, the larger one would be contained in the union.
So, by the proposition that any connected topological component is exactly any connected topological subspace that cannot be made larger, the union is a connected component of \(T\) in which \(T_j\) is contained.
