208: Union of Path-Connected Subspaces Is Path-Connected if Subspace of Point from Each Subspace Is Path-Connected

<The previous article in this series | The table of contents of this series | The next article in this series>

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
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

---

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 ${\cup }_{\alpha }{S}_{\alpha }$.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>