TITLE: How are a set of matrices a topological space?
QUESTION [8 upvotes]: For instance, is the topology on the set of all $2 \times 2$ real matrices basically $\mathbb{R}^4$

REPLY [7 votes]: There are many ways to define the topology on the space of matrices. 
For people studying riemannian geometry or lie groups, they usually identify $M_n(\mathbb{R})$ with $\mathbb{R}^{n\times n}$, so that they can use the Euclidean metric to introduce the metric tensor.
For operator theorists, the topology would follow from the operator norm
\begin{equation}\|A\|=sup\{\|Ax\|: x\in \mathbb{R}^n, \|x\|=1\},\end{equation} because this can be easily generalized to infinite dimensional spaces.
I also vaguely remember people doing numerical analysis have their own way to define a topology that help them measure error bounds in computations.
But as Nobert pointed out, on a finite dimensional vector space, there is only one natural topology in the sense that addition and scalar multiplication are continuous, so all these approaches give the essentially unique topology. This illustrates that topology is too 'sparse' sometimes and we need more rigid structures to control things due to different applications.