A description/proof of that continuous image of path-connected subspace of domain is path-connected on codomain
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 any restriction of any continuous map on the domain and the codomain is continuous.
Target Context
- The reader will have a description and a proof of the proposition that any continuous map image of any path-connected subspace of the domain is path-connected on the codomain.
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 spaces, \(T_1, T_2\), any continuous map, \(f: T_1 \to T_2\), and any path-connected subspace, \(T_3 \subseteq T_1\), the image, \(f (T_3)\), is path-connected on \(T_2\).
2: Proof
For any points, \(p_1, p_2 \in f (T_3)\), is there a path, \(\lambda: [0, r] \to f (T_3)\) such that \(p_1 = \lambda (0)\) and \(p_2 = \lambda (r)\)? We can take any point from \(f^{-1} (p_i) \cap T_3\) as \(p'_i\). As \(T_3\) is path-connected, there is a path, \(\lambda': [0, 1] \to T_3\), such that \(\lambda' (0) = p'_1\) and \(\lambda' (1) = p'_2\). Let us define \(\lambda:= f \vert_{T_3} \circ \lambda': [0, 1] \to T_3 \to f (T_3)\), which is continuous as a composition of continuous maps (\(f \vert_{T_3}\) is continuous by the proposition that any restriction of any continuous map on the domain and the codomain is continuous), and \(\lambda (0) = p_1\) and \(\lambda (1) = p_2\).
3: Note
If the domain is a path-connected topological space, the domain is a path-connected subspace of itself, so, the image of any path-connected topological space is path-connected.
