I have very little background in topology and while I do understand the definition, why do I see this space so often is something that you can't simply see from the definition itself.
My guess is that it is in some sense a generalisation of the spaces we deal with (say $\mathbb R^n$), which is general enough to include a variety of spaces, but restricted enough to keep the nice properties we want. Similar to, say, formulating results in analysis in a metric space (even if we're mostly interested in $\mathbb R^n$ or even $\mathbb R$), or probability results formulated in $\sigma$-finite spaces (even though we really have a finite space).
Therefore: is the guess above correct? If so, what are some of the nice properties? Is there a particular connection to probability theory?
I would consider answering the first question sufficient, but would very much welcome a context along the lines of the second and third question.