574: Affine Simplex Map into Finite-Dimensional Vectors Space Is Continuous w.r.t. Canonical Topologies

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description/proof of that affine simplex map into finite-dimensional vectors space is continuous w.r.t. canonical topologies

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Topics

About: vectors space
About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of affine simplex.
• The reader knows a definition of canonical topology for finite-dimensional real vectors space.
• The reader knows a definition of canonical $C^{\mathrm{\infty }}$ atlas for finite-dimensional real vectors space.
• The reader knows a definition of continuous map.
• The reader admits 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.

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Target Context

• The reader will have a description and a proof of the proposition that any affine simplex map into any finite-dimensional vectors space is continuous with respect to the canonical topologies of the domain and the codomain.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \{\text{ the }d\text{ -dimensional real vectors spaces }\}$ with the canonical topology
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the affine-independent sets of base points on }V\}$
$[p_0,...,p_n]$: $=\text{ the affine simplex }$
${\mathbb{R}}^{n+1}$: $=\text{ the Euclidean topological space }$
$T$: $=\{t=(t^0,...,t^n)\in {\mathbb{R}}^{n+1}|\sum _{j\in \{0,...,n\}}{t}^j=1\wedge 0\le t^j\}\subseteq {\mathbb{R}}^{n+1}$, as the topological subspace of ${\mathbb{R}}^{n+1}$
$f$: $:T\to V,t=(t^0,...,t^n)\mapsto \sum _{j\in \{0,...,n\}}{t}^j{p}_j$
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Statements:
$f\in \{\text{ the continuous maps }\}$
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2: Natural Language Description

For any $d$-dimensional real vectors space, $V$, with the canonical topology, any affine-independent set of base points on $V$, $\{p_0,...,p_n\}\subseteq V$, the affine simplex, $[p_0,...,p_n]$, the Euclidean topological space, ${\mathbb{R}}^{n+1}$, and $T:=\{t=(t^0,...,t^n)\in {\mathbb{R}}^{n+1}|\sum _{j\in \{0,...,n\}}{t}^j=1\wedge 0\le t^j\}\subseteq {\mathbb{R}}^{n+1}$, as the topological subspace of ${\mathbb{R}}^{n+1}$, the map, $f:T\to V,t=(t^0,...,t^n)\mapsto \sum _{j\in \{0,...,n\}}{t}^j{p}_j$, is continuous.

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3: Proof

Let us think of the Euclidean $C^{\mathrm{\infty }}$ manifold, ${\mathbb{R}}^{n+1}$. $T\subseteq {\mathbb{R}}^{n+1}$ is the topological subspace of the $C^{\mathrm{\infty }}$ manifold.

Let us think of the canonical $C^{\mathrm{\infty }}$ manifold, $V$, with the canonical $C^{\mathrm{\infty }}$ atlas induced from the Euclidean $C^{\mathrm{\infty }}$ manifold, ${\mathbb{R}}^d$. $V\subseteq V$ is the topological subspace of the $C^{\mathrm{\infty }}$ manifold.

$({\mathbb{R}}^{n+1},id)$ is a chart of the $C^{\mathrm{\infty }}$ manifold, ${\mathbb{R}}^{n+1}$.

$\{p_1-p_0,...,p_n-p_0\}$ is linearly independent on $V$, and we can take any basis, $\{p_1-p_0,...,p_n-p_0,b_{n+1},...,b_d\}$ for $V$. For any $v\in V$, $v=s^1(p_1-p_0)+...+s^n(p_n-p_0)+s^{n+1}{b}_{n+1}+...+s^d{b}_d$, and $(V,\varphi )$, $\varphi :V\to {\mathbb{R}}^d,v\mapsto (s^1,...,s^n,s^{n+1},...,s^d)$, is a chart of the canonical $C^{\mathrm{\infty }}$ manifold for $V$.

$f(t)=\sum _{j\in \{0,...,n\}}{t}^j{p}_j=\sum _{j\in \{0,...,n\}}{t}^j(p_j-p_0)+\sum _{j\in \{0,...,n\}}{t}^j{p}_0=\sum _{j\in \{1,...,n\}}{t}^j(p_j-p_0)+1p_0$.

So, let us take the map, $f^{\prime }:{\mathbb{R}}^{n+1}\to V,t\mapsto \sum _{j\in \{1,...,n\}}{t}^j(p_j-p_0)+1p_0$. $f^{\prime }{|}_{{\mathbb{R}}^{n+1}\cap T}=f{|}_{{\mathbb{R}}^{n+1}\cap T}$. The restricted coordinates function, $\varphi \circ f^{\prime }\circ {id}^{-1}{|}_{id({\mathbb{R}}^{n+1}\cap T)}:id({\mathbb{R}}^{n+1}\cap T)\to \varphi (V)$, is $(t^0,...,t^n)\mapsto (t^1+p_0^1,...,t^n+p_0^n,p_0^{n+1},...,p_0^d)$, where $\varphi (p_0)=(p_0^1,...,p_0^d)$, which is obviously continuous.

By 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, $f$ is continuous.

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References

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