definition of \(n \times n\) invertible real matrices topological group
Topics
About: group
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of \(m \times n\) real matrices topological space.
- The reader knows a definition of ring of \(n \times n\) ring matrices.
- The reader knows a definition of topological group.
Target Context
- The reader will have a definition of \(n \times n\) invertible real matrices topological group.
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:
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( M_n (\mathbb{R})\): \(= \text{ the matrices ring } \land = \text{ the matrices topological space }\)
\(*M_n (\mathbb{R})^{\times}\): \(= \{M \in M_n (\mathbb{R}) \vert \exists M^{- 1} \in M_n (\mathbb{R})\}\) as the topological subspace of \(M_n (\mathbb{R})\), \(\in \{\text{ the topological groups }\}\)
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Conditions:
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2: Note
Let us see that \(M_n (\mathbb{R})^{\times}\) is indeed a topological group.
\(M_n (\mathbb{R})^{\times}\) is a group, by the proposition that for any ring, the subset of the multiplicatively-invertible elements forms a multiplicative group.
\(M_n (\mathbb{R})\) is Hausdorff, because it is homeomorphic to \(\mathbb{R}^{n^2}\), which is Hausdorff.
\(M_n (\mathbb{R})^{\times}\) is Hausdorff, by the proposition that any subspace of any Hausdorff topological space is Hausdorff.
The multiplication, \(m: M_n (\mathbb{R})^{\times} \times M_n (\mathbb{R})^{\times} \to M_n (\mathbb{R})^{\times}\), is continuous, because it is a restriction of the multiplication on \(M_n (\mathbb{R})\) and the proposition that any restriction of any continuous map on the domain and the codomain is continuous applies: \(M_n (\mathbb{R})^{\times} \times M_n (\mathbb{R})^{\times}\) is a topological subspace of \(M_n (\mathbb{R}) \times M_n (\mathbb{R})\), by the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
Let us see that the inversion, \(i: M_n (\mathbb{R})^{\times} \to M_n (\mathbb{R})^{\times}\), is continuous.
There is the homeomorphism, \(f: M_n (\mathbb{R}) \to \mathbb{R}^{n^2}\), by which the topology of \(M_n (\mathbb{R})\) is defined.
The domain and codomain restriction, \(f^`: M_n (\mathbb{R})^{\times} \to f (M_n (\mathbb{R})^{\times})\), is a homeomorphism, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
\(i = {f^`}^{- 1} \circ f^` \circ i \circ {f^`}^{- 1} \circ f^`\).
Let us see that \(f^` \circ i \circ {f^`}^{- 1}: f (M_n (\mathbb{R})^{\times}) \to f (M_n (\mathbb{R})^{\times})\) is continuous.
\(f (M_n (\mathbb{R})^{\times}) \subseteq \mathbb{R}^{n^2}\) is a topological subspace of the \(C^{\infty}\) manifold, \(\mathbb{R}^{n^2}\), and by the standard chart for \(\mathbb{R}^{n^2}\), each component of each \(v \in f (M_n (\mathbb{R})^{\times})\) becomes a component of \({f^`}^{- 1} (v)\), each component of \(i ({f^`}^{- 1} (v))\) is a polynomial of the components of \({f^`}^{- 1} (v)\) divided by the determinant of \({f^`}^{- 1} (v)\), which is a polynomial of the components of \({f^`}^{- 1} (v)\), which is a polynomial of the components of \(v\) divided by a polynomial of the components of \(v\), and those components of \(i ({f^`}^{- 1} (v))\) become the components of \(f^` \circ i \circ {f^`}^{- 1} (v)\), so, the components of the components function of \(f^` \circ i \circ {f^`}^{- 1}\) are some polynomials divided by a polynomial, which are continuous.
So, \(f^` \circ i \circ {f^`}^{- 1}\) is 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.
\(f^`\) and \({f^`}^{- 1}\) are continuous, because \(f^`\) is a homeomorphism.
So, \(i = {f^`}^{- 1} \circ f^` \circ i \circ {f^`}^{- 1} \circ f^`\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
So, \(M_n (\mathbb{R})^{\times}\) is a topological group.
