130: Equivalence of Map Continuousness in Topological Sense and in Norm Sense for Coordinates Functions|T.B.P.

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A description/proof of equivalence of map continuousness in topological sense and in norm sense for coordinates functions

Topics

About: $C^{\infty}$ manifold

Starting Context

The reader knows a definition of $C^{\infty}$ manifold.
The reader knows a definition of continuous map.
The reader knows a definition of continuous, normed spaces map.

Target Context

The reader will have a description and a proof of 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.

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

C^{\infty}

manifolds,

M_{1}

and

M_{2}

, and any map,

f : M_{1} \to M_{2}

f

is continuous in the topological sense if and only if

f

is continuous in the norm sense for the coordinates functions.

2: Proof

Suppose that

f

is continuous in the norm sense for the coordinates functions. For any open set,

U_{2} \subseteq M_{2}

, around any point,

m_{2} \in U_{2}

, there is a chart,

(U_{m_{2}}, ϕ_{m_{2}}), ϕ_{m_{2}} : U_{m_{2}} \to ϕ_{m_{2}} (U_{m_{2}}) \subseteq R^{d_{2}}

where

U_{m_{2}} \subseteq U_{2}

f^{- 1} (m_{2})

may consist of multiple points, but around any point,

m_{1} \in f^{- 1} (m_{2})

, there is a chart,

(U_{m_{1}}, ϕ_{m_{1}}), ϕ_{m_{1}} : U_{m_{1}} \to ϕ_{m_{1}} (U_{m_{1}}) \subseteq R^{d_{1}}

, and by the continuousness in the norm sense, for any open ball,

B_{ϕ_{2} (m_{2}) - ϵ} \subseteq ϕ_{m_{2}} (U_{m_{2}})

, there is an open ball,

B_{ϕ_{1} (m_{1}) - δ} \subseteq ϕ_{m_{1}} (U_{m_{1}})

, such that

ϕ_{m_{2}} (f (ϕ_{m_{1}}^{- 1} (B_{ϕ_{1} (m_{1}) - δ}))) \subseteq B_{ϕ_{2} (m_{2}) - ϵ}

. But

ϕ_{m_{1}}^{- 1} (B_{ϕ_{1} (m_{1}) - δ})

, open on

M_{1}

, is included in

f^{- 1} (U_{2})

, because

f (ϕ_{m_{1}}^{- 1} (B_{ϕ_{1} (m_{1}) - δ})) \subseteq ϕ_{m_{2}}^{- 1} (B_{ϕ_{2} (m_{2}) - ϵ}) \subseteq U_{2}

. As

{m_{1}} = f^{- 1} (U_{2})

, there is an open set,

ϕ_{m_{1}}^{- 1} (B_{ϕ_{1} (m_{1}) - δ}) \subseteq f^{- 1} (U_{2})

, at any point,

m_{1} \in f^{- 1} (U_{2})

, so, by the local criterion for openness,

f^{- 1} (U_{2})

is open. So, as the preimage of any open set is open,

f

is continuous in the topological sense.

Suppose that

f

is continuous in the topological sense. For any point,

m_{1} \in M_{1}

where

f (m_{1}) = m_{2}

, there is a chart,

(U_{m_{2}}, ϕ_{m_{2}}), ϕ_{m_{2}} : U_{m_{2}} \to ϕ_{m_{2}} (U_{m_{2}}) \subseteq R^{d_{2}}

. There is an open ball,

B_{ϕ_{m_{2} - ϵ}} \subseteq ϕ_{m_{2}} (U_{m_{2}})

, and by the continuousness in the topological sense,

f^{- 1} (ϕ_{m_{2}}^{- 1} (B_{ϕ_{m_{2}} - ϵ}))

is open including

m_{1}

, so, there is a chart,

(U_{m_{1}}, ϕ_{m_{1}}), ϕ_{m_{1}} : U_{m_{1}} \to ϕ_{m_{1}} (U_{m_{1}}) \subseteq R^{d_{1}}

where

U_{m_{1}} \subseteq f^{- 1} (ϕ_{m_{2}}^{- 1} (B_{ϕ_{m_{2}} - ϵ}))

, and an open ball,

B_{ϕ_{m_{1}} - δ} \subseteq ϕ_{m_{1}} (U_{m_{1}})

, which satisfies

ϕ_{m_{2}} (f ({ϕ_{m_{1}}}^{- 1} (B_{ϕ_{m_{1}} - δ}))) \subseteq B_{ϕ_{m_{2}} - ϵ}

, because

B_{ϕ_{m_{1}} - δ}

is included in

ϕ_{m_{1}} (U_{m_{1}})

and

U_{m_{1}}

is included in

f^{- 1} (ϕ_{m_{2}}^{- 1} (B_{ϕ_{m_{2}} - ϵ}))

, which means that

f ({ϕ_{m_{1}}}^{- 1} (B_{ϕ_{m_{1}} - δ}))

is included in

{ϕ_{m_{2}}}^{- 1} (B_{ϕ_{m_{2}} - ϵ})

, which means that

ϕ_{m_{2}} (f ({ϕ_{m_{1}}}^{- 1} (B_{ϕ_{m_{1}} - δ})))

is included in

B_{ϕ_{m_{2}} - ϵ}

3: Note

While prevalently the continuousness of a map in topological sense is claimed by the continuousness of coordinates functions in norm sense, the argument is valid only because of this proposition or a like.

References

The Bias Planet
Definitions and Propositions
School Mathematics from Higher Viewpoints
To Disentangle Confusing Terms or Discourses
Information Tables
Exploiting LibreOffice or OpenOffice
Exposing the Java Programming Language
Exposing C++
Exposing Python
Exposing C#
Exposing Gradle
Exposing Git
UNO Dispatch Commands

2022-09-11