A definition of topological path-connected-ness of 2 points
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological path.
Target Context
- The reader will have a definition of path-connected-ness of any 2 points on any topological space.
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: Definition
For any topological space, \(T\), any points, \(p_1, p_2 \in T\), are path-connected if there is a path-connected topological subspace, \(T_1 \subseteq T\), that contains the 2 points
2: Note
Although there can be another, probably more plain, definition that requires just a path on \(T\) that connects the 2 points, it is equivalent to the definition of this article, 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.
The definition of this article is adopted in order to be more symmetric with the definition of topological connected-ness of 2 points.
