In the real vector space ${M}_2(\mathbb{R})$ of $2\times 2$ matrices with entries in $\mathbb{R}$, we define the topology such that the natural bijection from ${M}_2(\mathbb{R})$ to $\mathbb{R}^4$ is a homeomorphism. Is its linear subspace $$X:=\lbrace A\in M_2(\mathbb{R}):A^T A = I_2\rbrace$$ compact?
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Does this answer your question? https://math.stackexchange.com/questions/18653/compactness-of-the-set-of-n-times-n-orthogonal-matrices or this one https://math.stackexchange.com/questions/1422768/prove-that-the-set-of-orthogonal-matrices-is-compact?noredirect=1&lq=1 – Sumanta Jan 13 '22 at 16:42
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Please have a look at How to ask a good question. – WhatsUp Jan 13 '22 at 16:43
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yes it does, should I delete this answer? – Pablo Garcia Pastor Jan 13 '22 at 16:48
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@PabloGarciaPastor It's better to let the system close it as duplicate than to delete. Thanks for asking! – hunter Jan 13 '22 at 16:59