Intuition behind topological spaces

You’re much better off if you start with a known topological space, I suggest the real number system $\mathbb R$, and see how the axioms apply to it. Remember that a set $U$ of real numbers is open if $\forall x\in U$ there is $\varepsilon>0$ such that all the numbers within a distance of $\varepsilon$ from $x$ are in $U$. So your first task is to verify that an “open interval” such as $\langle0,1\rangle$ (the set of numbers strictly between $0$ and $1$) is open according to this definition. That is the beginning of your intuition. Now show that the “closed interval” $[0,1]$, namely the set of numbers between $0$ and $1$ including the endpoints, is not open. Now show that the intersection of two open sets is open. Now show that any union of open sets is open. Now show that the empty set is open. You have already shown that the whole set $\mathbb R$ is open, do you see why? Do you see why $0$ is “close” to $\langle0,1\rangle$ without being in this open set?

Your desire for a metric-free intuition is, in some sense, futile due to the fact that every topological space $(X,\tau )$ is metrizable as long as by metrizable you mean the existence of a value quantale $V$ (a certain axiomatization of some of the properties of $\mathbb R$) together with a function $d:X\times X\to V$ satisfying $d(x,x)=0$ and $d(x,z)\le d(x,y)+d(y,z)$, such that the open-ball topology for this metric is the original topology.

This is explained in detail in all topologies come from generalized metrics by Ralph Kopperman and Quantales and continuity spaces by Robbert Flagg.

You say that you don't want a "metric space" intuition, yet you want spaces to be "correctly spatially represented." I hate to say this, but all the phrases like "correctly spatially represented" and "nearby" are ideas of metric spaces. (For instance, you draw points close together in your pictures -- you're using the metric on your sheet of paper.) You won't get your non-metric intuitions using such thinking. So if you really demand some intuition that doesn't come from metric spaces, I suggest not using words like "nearness."

One intuition for open sets is that they are "fat" or "thick;" in fact, the "fattest" kinds of sets you can take in your given topological space.

A non-metric example of this intuition comes from the Zariski topology. If you don't know what this is, here is an illustration. Let $\mathbb C^n$ be the $n$-dimensional complex affine space. In the Zariski topology, a closed subset is a subset that is cut out by a collection of complex polynomials in the $n$ variables $z_1,\ldots,z_n$. For example, any singleton point is a closed subset (given by some equation $z_i = c_i$, where $c_i$ are the coordintes of the point) as are hypersurfaces given by some polynomial equation $f(\vec z) = 0$.

Note that the open ball (of say, radius $r < \infty$) in $\mathbb C^n$ is not an open set. (Its complement may be cut out by an inequality, but never by a complex polynomial.) In fact, this topology does not arise by putting a metric on $\mathbb C^n$. (And cannot; it's even non-Hausdorff.)

But its open sets are still "fat," in that open sets are always either empty, or complements of higher codimension subsets, so they are always top-dimension (i.e., n-complex dimensional) subsets. (By the way, if you are not comfortable thinking of the empty set as "fat," that's fine. All other open sets are "fat.")