A description/proof of that topological subspaces map continuousness at point is implied by continuousness of map 'open set'-wise extended to superspaces
Topics
About: topological space
About: map
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of continuous map at point.
- The reader admits the proposition that for any topological space and any sub topological space that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
Target Context
- The reader will have a description and a proof of the proposition that any map between any topological subspaces is continuous at any point if a map that is an extension of the original map to some open sets of the superspaces is continuous.
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 subspaces, \(T_{12} \subseteq T_{11}\) and \(T_{22} \subseteq T_{21}\), and any map, \(f: T_{12} \rightarrow T_{22}\), \(f\) is continuous at any point, \(p \in T_{12}\), if there is a continuous map, \(f': U'_p \rightarrow U'_{f (p)}\), where \(U'_p\) and \(U'_{f (p)}\) are a neighborhood of \(p\) on \(T_{11}\) and a neighborhood of \(f (p)\) on \(T_{21}\), such that \(f'|_{U'_p \cap T_{12}} = f\).
2: Proof
Let us suppose that there is such an \(f'\). For any neighborhood of \(f (p)\) on \(T_{22}\), \(U_{f (p)} \subseteq T_{22}\), there is an open set on \(T_{21}\), \(U'_1 \subseteq T_{21}\), such that \(U_{f (p)} = U'_1 \cap T_{22}\). \(U'_{f (p)} \cap U'_1\) is open on \(T_{21}\) and on \(U'_{f (p)}\), and \(f'^{-1} (U'_{f (p)} \cap U'_1)\) is open on \(U'_p\) and so, on \(T_{11}\) by the proposition that for any topological space and any sub topological space that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space. \(U_p := f'^{-1} (U'_{f (p)} \cap U'_1) \cap T_{12}\) is open on \(T_{12}\).
In fact, \(U_p\) is named so because \(p \in U_p\), because \(f' (p) = f (p) \in U'_{f (p)}\) and \(f' (p) = f (p) \in U_{f (p)} = U'_1 \cap T_{22}\), so, \(f' (p) \in U'_{f (p)} \cap U'_1\), so, \(p \in f'^{-1} (U'_{f (p)} \cap U'_1)\) and of course, \(p \in T_{12}\).
\(f (U_p) \subseteq U_{f (p)}\), because for any \(p_1 \in U_p\), \(f (p_1) = f' (p_1)\) because \(p_1 \in U'_p \cap T_{12}\) (as \(U_p\) is in a preimage under \(f'\)), so, \(f (p_1) \in U'_{f (p)} \cap U'_1\) (as \(U_p\) is in the preimage of \(U'_{f (p)} \cap U'_1\) under \(f'\)), but as \(f\) is into \(T_{22}\), \(f (p_1) \in T_{22}\), so, \(f (p_1) \in U'_{f (p)} \cap U'_1 \cap T_{22}\), but as \(U_{f (p)} = U'_1 \cap T_{22}\), \(f (p_1) \in U'_{f (p)} \cap U_{f (p)} \subseteq U_{f (p)}\).
So, for any neighborhood of \(f (p)\) on \(T_{22}\), \(U_{f (p)}\), there is a neighborhood of \(p\) on \(T_{12}\), \(U_p\), such that \(f (U_p) \subseteq U_{f (p)}\), which is nothing but the definition of continuousness at point.
3: Note
The superspaces are usually expected to be \(C^\infty\) manifolds, and the usual aim of bringing in the extended map is to check the continuousness of the original map with the extended map, which is expected to have a coordinates function representation, whose continuousness can be checked in norm sense, which equals the continuousness of the extended map, by the proposition that for any map between \(C^\infty\) manifolds, its continuousness in the topological sense equals its continuousness in the norm sense for the coordinates functions.
Note that the subspaces do not necessarily have any chart, because a subspace is not necessarily a \(C^\infty\) manifold and may be just a topological space, so, there may be no coordinates function representation of the map, by which the continuousness of the map could be checked.
Bringing in an arbitrary in-norm-sense continuous function between the topological subspaces, which (the function) is not based on the superspaces, should not guarantee the continuousness of the original map in topological sense, unless proved otherwise.
