265: Map Between Topological Spaces Is Continuous at Point if They Are Subspaces of C∞ 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

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that map between topological spaces is continuous at point if they are subspaces of $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ 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:
(
$\mathrm{\exists }{M}_1,M_2\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds }\}(T_1\subseteq M_1\wedge T_1\in \{\text{ the topological subspaces of }{M}_1\}\wedge T_2\subseteq M_2\wedge T_2\in \{\text{ the topological subspaces of }{M}_2\})$
$\wedge$
$\mathrm{\exists }(U_p^{\prime }\subseteq M_1,{\varphi }_p^{\prime })\in \{\text{ the charts around }p\}\wedge \mathrm{\exists }(U_{f(p)}^{\prime }\subseteq M_2,{\varphi }_{f(p)}^{\prime })\in \{\text{ the charts around }f(p)\}\wedge \mathrm{\exists }{f}^{\prime }:U_p^{\prime }\to U_{f(p)}^{\prime }(f^{\prime }{|}_{U_p^{\prime }\cap T_1}=f{|}_{U_p^{\prime }\cap T_1}\wedge {\varphi }_{f(p)}^{\prime }\circ f^{\prime }\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)}:{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)\to {\varphi }_{f(p)}^{\prime }(U_{f(p)}^{\prime })\in \{\text{ the continuous maps }\})$
)
$⟹$
$f\in \{\text{ the maps continuous at }p\}$
//

---

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 $C^{\mathrm{\infty }}$ manifold, $M_j$, and there are a chart, $(U_p^{\prime }\subseteq M_1,{\varphi }_p^{\prime })$, around $p$ and a chart, $(U_{f(p)}^{\prime }\subseteq M_2,{\varphi }_{f(p)}^{\prime })$, around $f(p)$, and a map, $f^{\prime }:U_p^{\prime }\to U_{f(p)}^{\prime }$, such that $f^{\prime }{|}_{U_p^{\prime }\cap T_1}=f{|}_{U_p^{\prime }\cap T_1}$, and the restricted coordinates function, ${\varphi }_{f(p)}^{\prime }\circ f^{\prime }\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)}:{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)\to {\varphi }_{f(p)}^{\prime }(U_{f(p)}^{\prime })$, is continuous.

---

3: Proof

As ${\varphi }_{f(p)}^{\prime }\circ f^{\prime }\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)}$ is continuous, $f^{\prime }{|}_{U_p^{\prime }\cap T_1}:U_p^{\prime }\cap T_1\to U_{f(p)}^{\prime }={{\varphi }_{f(p)}^{\prime }}^{-1}\circ {\varphi }_{f(p)}^{\prime }\circ f^{\prime }\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)}\circ {\varphi }_p^{\prime }{|}_{U_p^{\prime }\cap T_1}$ is continuous as a composition of continuous maps, because ${\varphi }_p^{\prime }{|}_{U_p^{\prime }\cap T_1}:U_p^{\prime }\cap T_1\to {\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)$ is a homeomorphism, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.

That means that $f{|}_{U_p^{\prime }\cap T_1}=f^{\prime }{|}_{U_p^{\prime }\cap T_1}:U_p^{\prime }\cap T_1\to U_{f(p)}^{\prime }$ is continuous, and $f{|}_{U_p^{\prime }\cap T_1}:U_p^{\prime }\cap T_1\to M_2$ 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 $f(p)$, $V_{f(p)}=V_{f(p)}^{\prime }\cap T_2$, where $V_{f(p)}^{\prime }\subseteq M_2$ is open on $M_2$. As $f{|}_{U_p^{\prime }\cap T_1}$ is continuous at $p$, there is an open subset, $V_p\subseteq U_p^{\prime }\cap T_1$, around $p$ such that $f{|}_{U_p^{\prime }\cap T_1}(V_p)\subseteq V_{f(p)}^{\prime }$. As $f$ is into $T_2$, $f{|}_{U_p^{\prime }\cap T_1}(V_p)\subseteq V_{f(p)}^{\prime }\cap T_2=V_{f(p)}$. As $V_p$ is open on $U_p^{\prime }\cap T_1$, $V_p=V_p^{\prime }\cap T_1$, where $V_p^{\prime }\subseteq U_p^{\prime }$, open on $U_p^{\prime }$. $V_p^{\prime }$ is open on $M_1$, by the proposition that any open set on any open topological subspace is open on the base space, so, $V_p=V_p^{\prime }\cap T_1$ is open on $T_1$. In fact, $f{|}_{U_p^{\prime }\cap T_1}(V_p)=f(V_p)$. 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 $C^{\mathrm{\infty }}$ 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 ${\varphi }_{f(p)}^{\prime }\circ f^{\prime }\circ {{\varphi }_p^{\prime }}^{-1}{|}_{{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)}:{\varphi }_p^{\prime }(U_p^{\prime }\cap T_1)\to {\varphi }_{f(p)}^{\prime }(U_{f(p)}^{\prime })=id\circ f^{\prime }\circ {id}^{-1}{|}_{id(M_1\cap T_1)}:id(M_1\cap T_1)\to id(M_2)=f^{\prime }{|}_{T_1}:T_1\to M_2$, 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 $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ manifolds; just sitting in the ambient Euclidean $C^{\mathrm{\infty }}$ manifolds sets-wise is not enough).

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>