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orthogonal matrix

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How can I show that O(n) and SO(n) x Z2 are homeomorphic? O(n) is the orthogonal group and SO(n) is the special orthogonal group.

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The determinant of an orthogonal matrix is provided.

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The determinant of an orthogonal matrix is either 1 or (-1). It's very easy to see. Since if A belongs to O(n), then we have A^T A = I, and
det(A) = det(A^T), taking the determinants of both sides of the equation A^T A = I, we get (det(A))^2 = 1, so det(A) = 1 or det(A) = -1.
The determinant of a matrix B from SO(n) ...

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