The question asks "Is the set of all $3\times3$ real invertible matrices connected?"

My intuitive idea is that we can establish a separation consisting of matrices with positive and negative determinant respectively, whose union will be the whole set but with no intersections. However I am not sure how to show intersection of one set with the closure of the other set is empty, according to the definition of being disconnected. (My guess is that the closure for real matrices with negative determinant is just itself union matrices with 0 determinant)

So what will be a rigorous proof of this, using only tools in point-set topology and knowledge in linear algebra?

I know that $3\times3$ matrices can be viewed as homeomorphic to $\mathbb{R}^9$, but how do we define the topology on this subspace (i.e. set of all $3\times3$ matrices)? What will an open set look like in this topology?