A description/proof of that 2 continuous maps from connected topological space into Hausdorff topological space such that, for any point, if they agree at point, they agree on neighborhood, totally agree or totally disagree
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of connected topological space.
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of continuous map.
- The reader knows a definition of neighborhood of point.
- The reader admits the proposition that any 2 continuous maps from any topological space into any Hausdorff topological space that (the maps) disagree at any point disagree on a neighborhood of the point.
- The reader admits the proposition that any 2 continuous maps from any connected topological space into any topological space such that, for any point, if they (the maps) agree at the point, they agree on a neighborhood and if disagree at the point they disagree on a neighborhood, totally agree or totally disagree on the whole domain.
Target Context
- The reader will have a description and a proof of the proposition that any 2 continuous maps from any connected topological space into any Hausdorff topological space such that, for any point, if they (the maps) agree at the point, they agree on a neighborhood, totally agree or totally disagree on the whole domain.
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 connected topological space, \(T_1\), any Hausdorff topological space, \(T_2\), and any continuous maps, \(f_1, f_2: T_1 \to T_2\), such that if \(f_1 (p) = f_2 (p)\) at any point, there is a neighborhood, \(N_p\), of \(p\) such that \(f_1 (p') = f_2 (p')\) for each \(p' \in N_p\), \(f_1 (p) = f_2 (p)\) for each \(p \in T_1\) or \(f_1 (p) \neq f_2 (p)\) for each \(p \in T_1\).
2: Proof
\(f_1\) and \(f_2\) disagree on a neighborhood if they disagree at any point, by the proposition that any 2 continuous maps from any topological space into any Hausdorff topological space that (the maps) disagree at any point disagree on a neighborhood of the point. So, \(f_1\) and \(f_2\) totally agree or totally disagree on whole \(T_1\), by the proposition that any 2 continuous maps from any connected topological space into any topological space such that, for any point, if they (the maps) agree at the point, they agree on a neighborhood and if disagree at the point they disagree on a neighborhood, totally agree or totally disagree on the whole domain.
3: Note
Any Euclidean topological space is Hausdorff, so, any 2 continuous maps from any connected topological space into any Euclidean topological space that (the maps) agree on a neighborhood if they agree at any point totally agree or totally disagree on the whole domain.
