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For instance, is the topology on the set of all $2 \times 2$ real matrices basically $\mathbb{R}^4$

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    Yes, exactly...2012-10-10
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    The topology is something you can *choose*. On $\mathbb{R}^n$ you usually *choose* the topology induced by the norm $||.||$. On the set of $2x2$ real matrices, there are lots of different ways to define a topology. Which topology are you interested it, i.e. what do you want to *do* with it?2012-10-10
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    That's one possible topology, and arguably the most natural. However there are lots of other, different, topologies you can put on your space. **Edit.** heh, @fgp got in a few seconds before my comment. Read that.2012-10-10

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