I am trying to check compactness connectedness ,path-connectedness ,open or closeness of the sets of all matrices of different types. For example set of symmetric matrices, set of skew symmetric matrices , set of orthogonal matrices , set of all invertible matrices over $\mathbb {R}$ etc. What type of methods should apply to find these properties of these sets. I tried one method for connectedness by checking whether it is convex as convex imply connected. Please suggest proper methods of these. If possible suggest some books related to these results.
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For some of these sets it is useful to consider their image under the determinant. – symplectomorphic Jun 01 '15 at 05:30
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How Det is a continuous function? – neelkanth Jun 01 '15 at 15:05
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@YOGESH As a polynomial function, say. – Did Jun 26 '15 at 05:34
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ok ...thanking you.. – neelkanth Jun 26 '15 at 05:54
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Firstly, to show such sets are closed/open, try considering the determinant function and functions like $f(A) = A^T - A$, and using the fact that the inverse image of a closed/open set by a continuous function is closed/open.
Then for compactness, try show that the set is bounded in an appropriate norm and use the fact that all norms induce the same topology for finite-dimensional vector spaces - we can think of the space of matrices $M_n(\mathbb{R})$ as isomorphic to $\mathbb{R}^{n^2}$.
For connectedness, you may want to try using the fact that the image of a (path) connected set under a continuous map is (path) connected. Also, what can you say about the connectedness properties of a finite-dimensional vector space?
I hope I have given you enough to get started. If not, the book on Lie Theory by Brian Hall may be helpful, or if you want more explicit hints, see the answers to this question on MSE.
