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A common idea in the nLab is to study an object by doing experiments on it. Mathematically, we study some object by either throwing some mud at it to see what sticks, or taking photographs of it and looking at the resulting pictures. Less prosaically, these tend to get called probes and coprobes. Completely concretely, we look at morphisms in to or out of our object from or to some family of known (and hopefully simple) spaces.
Then it is natural to ask “What does property X look like in experiments?”. I’d like to ask this in particular of compactness and I’m most interested in the case where the experiments start and end in $\mathbb{R}$.
(A fuller post follows in the first comment)
A common idea in the nLab is to study an object by doing experiments on it. Mathematically, we study some object by either throwing some mud at it to see what sticks, or taking photographs of it and looking at the resulting pictures. Less prosaically, these tend to get called probes and coprobes. Completely concretely, we look at morphisms in to or out of our object from or to some family of known (and hopefully simple) spaces.
Homotopy theory, cohomology theory, generalised smooth spaces; all of these are examples of this.
In studying Froelicher spaces, this idea is naturally at the forefront. So when studying the topology of a Froelicher space, I am most interested in topological notions that have this experimental feel. Some topological notions already have this built in (completely regular being an obvious one) whilst others are easily adaptable (normal springs to mind). Others may not be completely recastable in this form, but will have a “nearby” notion that is naturally of the experimental form; “Hausdorff”, for example, is difficult to cast precisely in a “maps out” form but one could talk of “functionally Hausdorff” meaning “can be separated by continuous functions (to, say, $\mathbb{R}$).
The topological property that I’m currently thinking about is compactness. Now, for “maps out” (“coprobes”), there’s an easy way to test compactness: is the image of a test function always compact? This may not pin down compactness precisely, but it’s obviously closely related (in fact, this is called “pseudocompact”).
For “maps in” (“probes”) there is a known test for compactness given by looking for limit points of directed sets. In terms of “maps in”, it asks “Can a map from a directed set be extended?”. The simplest case is when we restrict ourselves to $\mathbb{N}$ for the directed set and then we get sequential compactness.
However, coming from the realm of Froelicher spaces, I don’t want to use $\mathbb{N}$. I want to use $\mathbb{R}$. Topologically, therefore, I want some notion of “path-compact” that I can abstract to Froelicher spaces. (Note that here I’m interested in the topological situation; the Froelicher stuff is to explain my motivation.)
This may well be known, though I haven’t found a trace of it as yet, in which case please enlighten me! But if, as I suspect, it’s not, what should it look like? One can take this more generally, of course, since there is the notion of a “compact object” in an appropriate category (though I don’t know too much about that) and so one could ask a similar question in a suitable category: “Is there a compactness-like property that can be probed, given some fixed family of test objects from which to probe?”.
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