What's so discrete about discrete topology?

In the discrete topology, no two points are "near" each other (that is, there is always a neighborhood of one that does not contain the other). We think of the points as "highly separated" (as the integers are, as a subset of the reals). By contrast, the point $\{1\}$ is near the open interval $(0,1)$, in the usual topology of the reals, as every neighborhood of $1$ contains points of the interval.

Neighborhoods are, essentially, those points we consider "close" to a given point, and the finer the topology, the closer we come to being able to spatially distinguish distinct points. So the indiscrete topology is (by way of contrasting analogy), the space imagined as an indistinct "blob", whose points are all tightly bound together.

Topology is a way of thinking about proximity (which can be interpreted in many "non-standard" ways) without having the ability to rely on numbers to quantify this. It is a generalization of our "usual" way of quantifying closeness by distance (which form a special sub-class of topological spaces called metric spaces).

Intimately bound up with these notions of proximity, is the notion of connectedness, and discrete spaces are "highly disconnected". Intuitively, you cannot travel to point $a$, from point $b$ (if $a \neq b$), without exiting the space.

If $X$ is discrete, for each $x\in X$, $\{x\}$ is open. This means every element of $X$ has an open neighbourhood that contain only itself. Don't you find $X$ very discrete and separated?