A definition of path-connected topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of topological path.
Target Context
- The reader will have a definition of path-connected 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
Any topological space, T, such that for any points, \(p_1, p_2 \in T\), the points can be connected by a path, \(\lambda: [r_1, r_2] \rightarrow T\) where \([r_1, r_2]\) is any closed interval on \(\mathbb{R}\), as the terminal points, \(p_1 = \lambda (r_1)\) and \(p_2 = \lambda (r_2)\)
2: Note
It has to be a path, not just a curve.
'Path-connected' is always as a topological space, not as a subset on a topological space, which means that when a subset is said to be path-connected, it is as the topological subspace, which means that it is not about whether there is a path that is continuous as a map to the ambient topological space, but about whether there is a path that is continuous as a map to the subspace. But in fact, if the path is continuous as on the ambient space, it is continuous as on the subspace by the proposition that any restriction of any continuous map on the domain and the codomain is continuous and if the path is continuous as on the subspace, it is continuous as on the ambient space by the proposition that any expansion of any continuous map on the codomain is continuous, so, the distinction does not really matter.
We can demand \([r_1, r_2]\) to be \([0, 1]\) or whatever \([r_3, r_4]\) where \(r_3 \neq r_4\), because if there is a \(\lambda: [r_1, r_2] \rightarrow T\), there is the \({\lambda}': [r_3, r_4] \rightarrow [r_1, r_2] \rightarrow T\) where \(f: [r_3, r_4] \rightarrow [r_1, r_2]\) is \(r \mapsto r_1 + \frac{r_2 - r_1}{r_4 - r_3} (r - r_3)\).
