A description/proof of that 2 points that are path-connected on topological subspace are path-connected on larger subspace
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological path-connected-ness of 2 points.
- The reader admits the proposition that any 2 points are path-connected on any topological space if and only if there is a path that connects the 2 points on the topological space.
- The reader admits the proposition that any expansion of any continuous map on the codomain is continuous.
Target Context
- The reader will have a description and a proof of the proposition that any 2 points that are path-connected on any topological subspace are path-connected on any larger subspace.
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, \(T\), any subspace, \(T_1 \subseteq T\), and any 2 points, \(p_1, p_2 \in T_1\), that are path-connected on \(T_1\), \(p_1\) and \(p_2\) are path-connected on any larger subspace, \(T_2\), such that \(T_1 \subseteq T_2 \subseteq T\).
2: Proof
By the proposition that any 2 points are path-connected on any topological space if and only if there is a path that connects the 2 points on the topological space, there is a path, \(\lambda: [0, 1] \rightarrow T_1\) such that \(\lambda (0) = p_1\) and \(\lambda (1) = p_2\). \(\lambda ([0, 1]) \subseteq T_1 \subseteq T_2\). \({\lambda}': [0, 1] \rightarrow T_2\) as the expansion of \(\lambda\) on the codimension is a path on \(T_2\), by the proposition that any expansion of any continuous map on the codomain is continuous. By the proposition that any 2 points are path-connected on any topological space if and only if there is a path that connects the 2 points on the topological space, \(p_1\) and \(p_2\) are path-connected on \(T_2\).
