description/proof of that affine simplex map into finite-dimensional vectors space is continuous w.r.t. canonical topologies
Topics
About: vectors space
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
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 \(C^\infty\) 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 \(C^\infty\) 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.
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.
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:
\(V\): \(\in \{\text{ the } d \text{ -dimensional real vectors spaces }\}\) with the canonical topology
\(\{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\([p_0, ..., p_n]\): \(= \text{ the affine simplex }\)
\(\mathbb{R}^{n + 1}\): \(= \text{ the Euclidean topological space }\)
\(T\): \(= \{t = (t^0, ..., t^n) \in \mathbb{R}^{n + 1} \vert \sum_{j \in \{0, ..., n\}} t^j = 1 \land 0 \le t^j\} \subseteq \mathbb{R}^{n + 1}\), 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 \(T := \{t = (t^0, ..., t^n) \in \mathbb{R}^{n + 1} \vert \sum_{j \in \{0, ..., n\}} t^j = 1 \land 0 \le t^j\} \subseteq \mathbb{R}^{n + 1}\), 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.
3: Proof
Let us think of the Euclidean \(C^\infty\) manifold, \(\mathbb{R}^{n + 1}\). \(T \subseteq \mathbb{R}^{n + 1}\) is the topological subspace of the \(C^\infty\) manifold.
Let us think of the canonical \(C^\infty\) manifold, \(V\), with the canonical \(C^\infty\) atlas induced from the Euclidean \(C^\infty\) manifold, \(\mathbb{R}^d\). \(V \subseteq V\) is the topological subspace of the \(C^\infty\) manifold.
\((\mathbb{R}^{n + 1}, id)\) is a chart of the \(C^\infty\) 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 \((V, \phi)\), \(\phi: V \to \mathbb{R}^d, v \mapsto (s^1, ..., s^n, s^{n + 1}, ..., s^d)\), is a chart of the canonical \(C^\infty\) manifold for \(V\).
\(f (t) = \sum_{j \in \{0, ..., n\}} t^j p_j = \sum_{j \in \{0, ..., n\}} t^j (p_j - p_0) + \sum_{j \in \{0, ..., n\}} t^j p_0 = \sum_{j \in \{1, ..., n\}} t^j (p_j - p_0) + 1 p_0\).
So, let us take the map, \(f': \mathbb{R}^{n + 1} \to V, t \mapsto \sum_{j \in \{1, ..., n\}} t^j (p_j - p_0) + 1 p_0\). \(f'\vert_{\mathbb{R}^{n + 1} \cap T} = f\vert_{\mathbb{R}^{n + 1} \cap T}\). The restricted coordinates function, \(\phi \circ f' \circ {id}^{-1}\vert_{id (\mathbb{R}^{n + 1} \cap T)}: id (\mathbb{R}^{n + 1} \cap T) \to \phi (V)\), is \((t^0, ..., t^n) \mapsto (t^1 + p^1_0, ..., t^n + p^n_0, p^{n + 1}_0, ..., p^d_0)\), where \(\phi (p_0) = (p^1_0, ..., p^d_0)\), 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 \(C^\infty\) 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.
