description/proof of that map between topological spaces is continuous at point if they are subspaces of \(C^\infty\) manifolds and there are charts of manifolds around point and point image and map between chart open subsets which is restricted to original map whose restricted coordinates function is continuous
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
- 4: Note
Starting Context
- The reader knows a definition of continuous map at point.
- The reader knows a definition of \(C^\infty\) manifold.
- The reader knows a definition of subspace topology.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
- The reader admits the proposition that any expansion of any continuous map on the codomain is continuous.
- The reader admits the proposition that any open set on any open topological subspace is open on the base space.
Target Context
- The reader will have a description and a proof of the proposition that any topological spaces map is continuous at any point if the spaces are the subspaces of some \(C^\infty\) manifolds and there are some charts of the manifolds around the point and the point image and a map between the chart open subsets which (the map) is restricted to the original map whose (the chart open subsets map's) coordinates function's restriction 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(: T_1 \to T_2\)
\(p\): \(\in T_1\)
//
Statements:
(
\(\exists M_1, M_2 \in \{\text{ the } C^\infty \text{ manifolds }\} (T_1 \subseteq M_1 \land T_1 \in \{\text{ the topological subspaces of } M_1\} \land T_2 \subseteq M_2 \land T_2 \in \{\text{ the topological subspaces of } M_2\})\)
\(\land\)
\(\exists (U'_p \subseteq M_1, \phi'_p) \in \{\text{ the charts around } p\} \land \exists (U'_{f (p)} \subseteq M_2, \phi'_{f (p)}) \in \{\text{ the charts around } f (p)\} \land \exists f': U'_p \to U'_{f (p)} (f' \vert_{U'_p \cap T_1} = f \vert_{U'_p \cap T_1} \land \phi'_{f (p)} \circ f' \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap T_1)}: \phi'_p (U'_p \cap T_1) \to \phi'_{f (p)} (U'_{f (p)}) \in \{\text{ the continuous maps }\})\)
)
\(\implies\)
\(f \in \{\text{ the maps continuous at } p\}\)
//
2: Natural Language Description
For any topological spaces, \(T_1, T_2\), any map, \(f: T_1 \to T_2\), and any point, \(p \in T_1\), \(f\) is continuous at \(p\), if \(T_j \subseteq M_j\) is the subspace of a \(C^\infty\) manifold, \(M_j\), and there are a chart, \((U'_p \subseteq M_1, \phi'_p)\), around \(p\) and a chart, \((U'_{f (p)} \subseteq M_2, \phi'_{f (p)})\), around \(f (p)\), and a map, \(f': U'_p \to U'_{f (p)}\), such that \(f' \vert_{U'_p \cap T_1} = f \vert_{U'_p \cap T_1}\), and the restricted coordinates function, \(\phi'_{f (p)} \circ f' \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap T_1)}: \phi'_p (U'_p \cap T_1) \to \phi'_{f (p)} (U'_{f (p)})\), is continuous.
3: Proof
As \(\phi'_{f (p)} \circ f' \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap T_1)}\) is continuous, \(f' \vert_{U'_p \cap T_1}: U'_p \cap T_1 \to U'_{f (p)} = {\phi'_{f (p)}}^{-1} \circ \phi'_{f (p)} \circ f' \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap T_1)} \circ \phi'_p \vert_{U'_p \cap T_1}\) is continuous as a composition of continuous maps, because \(\phi'_p \vert_{U'_p \cap T_1}: U'_p \cap T_1 \to \phi'_p (U'_p \cap T_1)\) is a homeomorphism, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
That means that \(f \vert_{U'_p \cap T_1} = f' \vert_{U'_p \cap T_1}: U'_p \cap T_1 \to U'_{f (p)}\) is continuous, and \(f \vert_{U'_p \cap T_1}: U'_p \cap T_1 \to M_2\) is continuous, by the proposition that any expansion of any continuous map on the codomain is continuous.
For any open subset, \(V_{f (p)} \subseteq T_2\), around \(f (p)\), \(V_{f (p)} = V'_{f (p)} \cap T_2\), where \(V'_{f (p)} \subseteq M_2\) is open on \(M_2\). As \(f \vert_{U'_p \cap T_1}\) is continuous at \(p\), there is an open subset, \(V_p \subseteq U'_p \cap T_1\), around \(p\) such that \(f \vert_{U'_p \cap T_1} (V_p) \subseteq V'_{f (p)}\). As \(f\) is into \(T_2\), \(f \vert_{U'_p \cap T_1} (V_p) \subseteq V'_{f (p)} \cap T_2 = V_{f (p)}\). As \(V_p\) is open on \(U'_p \cap T_1\), \(V_p = V'_p \cap T_1\), where \(V'_p \subseteq U'_p\), open on \(U'_p\). \(V'_p\) is open on \(M_1\), by the proposition that any open set on any open topological subspace is open on the base space, so, \(V_p = V'_p \cap T_1\) is open on \(T_1\). In fact, \(f \vert_{U'_p \cap T_1} (V_p) = f (V_p)\). So, for any open set, \(V_{f (p)} \subseteq T_2\), around \(f (p)\), there is an open set, \(V_p \subseteq T_1\), around \(p\) such that \(f (V_p) \subseteq V_{f (p)}\), which means that \(f\) is continuous at \(p\).
4: Note
\(T_1, T_2\) have to be the topological subspaces of \(M_1, M_2\): just being in \(M_1, M_2\) set-wise does not make them topological subspaces.
This proposition is typically used when \(T_1, T_2\) are some subspaces of some Euclidean \(C^\infty\) manifolds, \(M_1, M_2\), and \(f\) is expressed with the standard coordinates of \(M_1\) and \(M_2\). Then, the expression is in fact \(\phi'_{f (p)} \circ f' \circ {\phi'_p}^{-1} \vert_{\phi'_p (U'_p \cap T_1)}: \phi'_p (U'_p \cap T_1) \to \phi'_{f (p)} (U'_{f (p)}) = id \circ f' \circ {id}^{-1} \vert_{id (M_1 \cap T_1)}: id (M_1 \cap T_1) \to id (M_2) = f' \vert_{T_1}: T_1 \to M_2\), and if that is continuous, \(f\) will be continuous.
While some textbooks nonchalantly claim the continuousness of \(f\) by expressing \(f\) with the coordinates of some ambient Euclidean \(C^\infty\) manifolds (How does the expression with the coordinates of the ambient spaces (not of \(T_1\) and \(T_2\)) validate the claim?), this proposition validates the claim (\(T_1\) and \(T_2\) have to be the topological subspaces of the ambient Euclidean \(C^\infty\) manifolds; just sitting in the ambient Euclidean \(C^\infty\) manifolds sets-wise is not enough).
