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 $X$.
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.