How to Axiomize the Notion of "Continuous Space"?

There was a teacher of mine which used to restrict the definition of continuity to the points of the set which were limit points. Hence, asking if the function $f:\{0 \}\cup [1,2] \to \mathbb{R}$ was continuous was not meaningful to him.

I asked him "why are you restricting it?" He answered (paraphrasing, of course. I don't remember the exact words): "Because continuity is a concept related to the process of limits, it is not quite meaningful to attribute the word 'continuous' to a point to which nothing is converging to." This phrase is quite, quite similar to your "proper axiomatization of a continuous space", whatever that means.

Turns out that later in class he needed to use the continuity of a function which he was restricting. I asked promptly: "Why is this function continuous? You first need to prove that all points of your set are limit points." After some discussion, he conceded on his definition of continuity.

The bottom line is: A lot of times in mathematics, it so happens that grasping to one's psychological comfort only leads to unnecessary hindrance. Making more assumptions than needed will often lead to useless labor, and also to wasted time.

If you didn't allow these two sorts of topology (discrete and trivial,) then if $X$ was a space with topology $\tau$, and $Y\subset X$, you couldn't always define a topology on $Y$.

The most obvious example is that the discrete topology on $\mathbb Z$ is the topology you get by considering $\mathbb Z\subset \mathbb R$ with the usual topology on $\mathbb R$.

It's a little harder to get trivial topologies, since you'd have to start with a non-Hausdorff topology.

I'm guessing there are category-theory reasons, also - that limits or co-limits might fail to exist in the category of topological spaces. But that's possibly more advanced than you need, and I'm not sure it is true.

The original definitions of topology had pretty strong axioms, to match metric spaces more closely.

But then people encountered "spaces" where fewer and fewer of the separation rules were satisfied. But I'm not sure I've ever personally seen a topology used in real math that was not $T_0$.

This question is way too tl;dr for me, but one phrase that hasn't been thrown around here, as far as I can tell, is "uniform space," and I wonder if that's one way of getting at what you want from a notion of "continuous space," since it is the topological setting were Cauchy completeness makes sense -- one step up in generality from metric spaces. (See also the nLab page.)

Leaving this as an answer rather than a comment since there are already so many comments.