A description/proof of that union of path-connected subspaces is path-connected if subspace of point from each subspace is path-connected
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of path-connected topological space.
- The reader admits the proposition that topological path-connected-ness of 2 points is an equivalence relation.
- The reader admits the proposition that any 2 points that are path-connected on any topological subspace are path-connected on any larger subspace.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space, the union of any possibly uncountable number of path-connected subspaces is path-connected if the subspace of a point from each subspace is path-connected.
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: Description
For any topological space, \(T\), and any possibly uncountable number of path-connected topological subspaces, \(\{S_\alpha\}\), \(S_\alpha \subseteq T\), \(\cup_\alpha S_\alpha\) is path-connected if the subspace as a point from each \(S_\alpha\), \(\{p_\alpha\}\) where \(p_\alpha \in S_\alpha\), is path-connected.
2: Proof
Note the proposition that any 2 points that are path-connected on any topological subspace are path-connected on any larger subspace.
For any \(p_1, p_2 \in \cup_\alpha S_\alpha\), \(p_1 \in S_{\alpha_1}\) and \(p_2 \in S_{\alpha_2}\) for an \(\alpha_1\) and an \(\alpha_2\). \(p_1\) and \(p_{\alpha_1}\) are path-connected on \(S_{\alpha_1}\), so, path-connected on the larger \(\cup_\alpha S_\alpha\), \(p_2\) and \(p_{\alpha_2}\) are path-connected on \(S_{\alpha_2}\), so, path-connected on the larger \(\cup_\alpha S_\alpha\), and \(p_{\alpha_1}\) and \(p_{\alpha_2}\) are path-connected on \(\{p_\alpha\}\), so, path-connected on the larger \(\cup_\alpha S_\alpha\), so, by the proposition that path-connected-ness of 2 points is an equivalence relation, \(p_1\) and \(p_2\) are path-connected on \(\cup_\alpha S_\alpha\).
