There are a myriad different variations on the theme of "compactness", and some of them have even made it on to Wikipedia. I'm interested in finding out more about types of compactness that fit the concept of doing experiments on an object to find out about it.

Let me explain the idea in a little more detail. In some branches of mathematics, we try not to study an object itself too closely. Category theory could be viewed as the extreme example of this, but it also finds a home in homotopy theory, cohomology theory, differential topology and geometry, and no doubt in others as well. The idea being that one has a family of "nice" objects and uses them to "probe" or "coprobe" the object in question. More prosaically, one tries to figure out the shape of the object in question either by throwing mud at it (to see what sticks) or by taking photographs of it. Less prosaically, we study $X$ by looking at morphisms $g \colon Y \to X$ or $f \colon X \to Y$ (where $Y$ runs over our family of "nice" objects).

For some property of our object, we can then ask "Can we detect it by doing these experiments?", or, more generally, "Is there something a bit like it that I can detect using experiments?" (the belief that these are the same question could go on the "false mathematical beliefs" question!).

Let's home in on the case I'm interested in: compactness. If we were probing our space by looking at maps from $\mathbb{N}$ (aka sequences), then we would come up with the notion of "sequentially compact". If we were probing our space by looking at maps to $\mathbb{R}$ (aka functionals), then we would come up with the notion of "pseudocompactness" (no, I'd never heard of it either before I looked it up on Wikipedia a couple of days ago!).

So, to my question: are there other examples of variations on the theme of compactness that fit this pattern? I'm particularly interested in the case of maps from $\mathbb{R}$ (path-compactness, perchance?) but anything of this flavour would be helpful.

I'm specifically looking for stuff that's known. If there's nothing that's known then I'm interested in trying to figure out what it should look like, but that's not a good MO question so I'm not asking it. Anyone interested in this wider question is welcome to join in a discussion on it over at the nForum.