This is an attempt to synthesize ideas that have appeared in other answers, for example sigfpe's and Tim Perutz's. Feel free to edit if you think the ideas can be better expressed.
The idea I want to back is that a topological space is an environment $X$ in which the notion of checking the truth of a statement locally makes sense. In the actual language of topological spaces, we want to be able to talk about statements which are true for a space $X$ if and only if they're true for every open set in an open cover of $X$, and the same should be true for every subspace of $X$. (For example, continuity and differentiability of a function both have this property.)
But whatever an open cover is, it should consist of elements chosen from a distinguished collection of subsets $\mathcal{P}$ of $X$ having certain properties. The empty set and $X$ should both be in $\mathcal{P}$ because checking a statement about $X$ is trivially equivalent to checking it on $X$ and on the empty set. $\mathcal{P}$ should be closed under arbitrary unions because a collection of open sets automatically forms an open cover of its union. $\mathcal{P}$ should be closed under binary intersections because one should be able to build an open cover of a subspace $S$ of $X$ by intersecting an open cover of $X$ with $S$, and if $S$ is itself open, an open cover of $S$ should be extendable to an open cover of $X$.
I don't think I've explained myself very well, though. I also wish I knew enough to say something about the relationship between topology and logic that the above seems to suggest. But one reason I like this perspective is that it suggests certain definitions naturally, such as the definition of compactness or of a manifold.
Some soapboxing: while I can see the pedagogical value of thinking about topological spaces as a natural generalization of metric spaces or even just of $\mathbb{R}$, I think the idea of a topological space is deeper than these roots suggest and I think Minhyong is looking for an answer that reflects this. In other words, I am of the opinion that the definition of a topological space is more natural than the definition of a metric space (or even of $\mathbb{R}$!), so one shouldn't use the latter to motivate the former. But this is just an opinion.