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**oblixps** how do you show that M_n(R) and R^(n^2) are homeomorphic? i defined a function f:M_n(R) -> R^(n^2) such that if you input any real matrix, you get out a vector that is set up as (a11, a12, ... a21, a22, ..., a_nn) so basically i go along the first row then the second row then 3rd row and so on of the matrix and put the elements in the order i get to them while doing this process. so i can see that this function is a bijection and now i need to show that both the function f and f^-1 are continuous and i am a bit lost on how to do that.

I don't understand the problem here? I assume that you're endowing $\displaystyle \text{Mat}_n\left(\mathbb{R}\right)$ with the topology induced by the usual matrix norm $\displaystyle \displaystyle |||A|||=\sqrt{\sum_{i=1}^{n}\sum_{j=1}|a_{i,j}|^2} ,\quad A=[a_{i,j}]$. If so, isn't

$\displaystyle \displaystyle f:\text{Mat}_n\left(\mathbb{R}\right)\to\mathbb{R} ^{n^2}:\begin{pmatrix}a_{1,1} & \cdots & a_{1,n}\\ \vdots & \ddots & \vdots\\ a_{n,1} & \cdots & a_{n,n}\end{pmatrix}\mapsto \left(a_{1,1},\cdots,a_{n,1},\cdots,a_{1,n} & \cdots a_{n,n}\right)$

a surjective isometry, and thus trivially a homeomorphism?