575: Affine Simplex on Finite-Dimensional Real Vectors Space Is Closed and Compact on Canonical Topological Superspace

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description/proof of that affine simplex on finite-dimensional real vectors space is closed and compact on canonical topological superspace

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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 closed set.
• The reader knows a definition of compact subset of topological space.
• The reader knows a definition of continuous map.
• The reader admits 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.
• The reader admits the proposition that the domain of any affine simplex map is closed and compact on the Euclidean topological superspace.
• The reader admits the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is compact.
• The reader admits the proposition that any compact subset of any Hausdorff topological space is closed.

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Target Context

• The reader will have a description and a proof of the proposition that any affine simplex on any finite-dimensional real vectors space is closed and compact on the canonical topological superspace.

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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
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the affine-independent sets of base points on }V\}$
$[p_0,...,p_n]$: $=\text{ the affine simplex }$
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Statements:
$[p_0,...,p_n]$ is closed and compact on $V$.
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$[p_0,...,p_n]$ is a compact topological space by itself, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.

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2: Natural Language Description

For any $d$-dimensional real vectors space, $V$, with the canonical topology, and any affine-independent set of base points on $V$, $\{p_0,...,p_n\}\subseteq V$, the affine simplex, $[p_0,...,p_n]$, is closed and compact on $V$.

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3: Proof

For the affine simplex map, $f:T\to V,t=(t^0,...,t^n)\mapsto \sum _{j\in \{0,...,n\}}{t}^j{p}_j$, where $T:=\{t=(t^0,...,t^n)\in {\mathbb{R}}^{n+1}|\sum _{j\in \{0,...,n\}}{t}^j=1\wedge 0\le t^j\}\subseteq {\mathbb{R}}^{n+1}$ as the subspace of ${\mathbb{R}}^{n+1}$, $f$ is continuous, by 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, and $T$ is compact, by the proposition that the domain of any affine simplex map is closed and compact on the Euclidean topological superspace.

So, $[p_0,...,p_n]$ is compact on $V$, by the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is compact.

As $V$ is a Hausdorff topological space (because $V$ is homeomorphic to ${\mathbb{R}}^d$ and ${\mathbb{R}}^d$ is Hausdorff), $[p_0,...,p_n]$ is closed on $V$, by the proposition that any compact subset of any Hausdorff topological space is closed.

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References

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