Exercise 1963Topology › Topology of Mn(ℝ)L1Added 4 Nov 2026

Classical closed subgroups of $GL_n(\mathbb{R})$

#closed subgroups#GL_n#SL_n#O_n#determinant

Statement

Show that the following subsets of $M_n(\mathbb{R})$ are closed:
$SL_n(\mathbb{R})=\{A:\det A=1\},\qquad O_n(\mathbb{R})=\{A:A^TA=I_n\},\qquad S_n(\mathbb{R})=\{A:A^T=A\}.$

Hint

Each is the preimage of a closed set under a continuous map.

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Solution

$SL_n$ . The determinant $\det:M_n(\mathbb{R})\to\mathbb{R}$ is continuous (polynomial in the entries). $SL_n=\det^{-1}(\{1\})$ is closed as preimage of the closed set $\{1\}$ .

$O_n$ . The map $\Phi(A)=A^TA-I_n$ is continuous into the symmetric matrices. $O_n=\Phi^{-1}(\{0\})$ is closed.

$S_n$ . The map $A\mapsto A^T-A$ is linear (continuous). $S_n$ is its kernel, hence closed.

Conclusion. $SL_n$ , $O_n$ , $S_n$ are closed subgroups (or subspace) of $M_n(\mathbb{R})$ .

Classical closed subgroups of $GL_n(\mathbb{R})$

Statement

Solution

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