# The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$

Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $$\mathbf{Top}$$ in order to freely use exponential laws for mapping spaces, which imply things like "the internal categorical product of quotient maps is a quotient map." I have a situation where an argument works regardless of the particular "convenient" subcategory that I pick. Hence, as long as a space lies in some coreflective Cartesian closed subcategory, I may apply my argument.

What is the class of topological spaces, that lie in some coreflective Cartesian closed subcategory of $$\mathbf{Top}$$?

Update: It is well-known that every class of spaces generates a Cartesian closed subcategory of $$\mathbf{Top}$$ (See Booth-Tillotson) but it may not be coreflective. On the other hand, many coreflective categories like the category of locally path-connected spaces are not Cartesian closed. As pointed out by David White below, it is a result of Juraj Cincura that there is no largest coreflective Cartesian closed subcategory of $$\mathbf{Top}$$.

I'd be content to just to know that there is a space that does not lie in any coreflective Cartesian closed subcategory of $$\mathbf{Top}$$.

• I'm accepting David's answer since it helped lead to a suitable conclusion. – Jeremy Brazas Apr 2 at 2:24

The relevant paper is Cartesian closed coreflective subcategories of the category of topological spaces by Juraj Cincura. The first line of the abstract says "Answering the first part of Problem 7 in [10] we prove that there is no largest Cartesian closed coreflective subcategory of the category Top of topological spaces and the same result for the category Haus of Hausdorff spaces."

Another great paper, by Vogt, is Convenient Categories of Topological Spaces for Homotopy Theory. It gives several examples, and argues the pros/cons of each. The author suggests that the coreflective subcategory generated by the locally compact spaces is large enough to contain everything he is interested in.

• Thanks David! Cincura's paper may be the closest thing out there. I'm mostly interested in metrizable spaces and could live happily in most convenient categories. My curiosity is whether this is necessary at all (for my particular situation). My guess is that there is some space that is not an object in any ccc subcategory. – Jeremy Brazas Mar 28 at 13:44
• But, look at the paper of Vogt. You can apply it to any small full subcategory of Top. So, if there was a space like the one you are thinking of, couldn't you just make the ccc category generated by it? You'd need to work with a (e.g. the smallest) subcategory satisfying Vogt's Axiom 2 and containing your space. Then Vogt's paper shows that the resulting category generated by this is ccc (not sure if you also need to close up under his Axiom 1, as I haven't thought about this paper in a while). – David White Mar 28 at 13:58
• Vogt's approach also shows up in slightly more general results by O. Wyler [sciencedirect.com/science/article/pii/0016660X72900141] and Booth-Tillotson [projecteuclid.org/euclid.pjm/1102779712]. One can take any class of spaces and turn it into a cartesian closed category but it won't be coreflective. The only way to make it coreflective (closed under sums and quotients so that colimits work out nicely) that I've ever seen requires compactness assumptions about the class you're working with. – Jeremy Brazas Mar 28 at 15:03
• Some non-compact spaces like $\mathbb{Q}$ and $\mathbb{R}$ still generate convenient categories (sequential spaces and delta-generated spaces respectively) but it's unclear to me how far this goes. – Jeremy Brazas Mar 28 at 15:09
• Given an arbitrary space X, suppose you take the 1-point compactification of X, then the ccc subcategory generated by it under Vogt's axioms. I think Vogt has a result that says, under his axioms, that open subsets are objects of the subcategory, so X would be. This would answer your 2nd question. I'm in an airport about to board, so no time to check the details here, but I wanted to share the idea. – David White Mar 31 at 12:09

I contacted Juraj Činčura and he kindly wrote back and directed me to the following observation that is a consequence of results in the paper that David White noted in his answer.

Cartesian closed coreflective subcategories of the category of topological spaces, Topology and its Applications 41 (1991) 205-212.

For an infinite cardinal $$a$$, let $$C(a)$$ be the space defined on set $$a\cup \{a\}$$ where $$V\subseteq C(a)$$ is open if and only if $$V\subseteq a$$ or if $$a\in V$$ and $$|a\backslash V|.

Činčura's Proposition 2.1 states that the coreflective hull of $$C(a)$$ in $$\mathbf{Top}$$ or $$\mathbf{Haus}$$ is Cartesian closed. However, Proposition 3.1 states that if $$a$$ is a strictly larger cardinal than $$b$$ and class $$\mathscr{B}$$ contains both $$C(a)$$ and $$C(b)$$, then the coreflective hull of $$\mathscr{B}$$ in $$\mathbf{Top}$$ or $$\mathbf{Haus}$$ is not Cartesian closed. Since $$C(a)$$ and $$C(b)$$ are both quotients of $$C(a)\times C(b)$$, this implies that if $$a>b$$, then any coreflective subcategory in $$\mathbf{Top}$$ or $$\mathbf{Haus}$$ containing the space $$C(a)\times C(b)$$ is not Cartesian closed.

Hence, there is no coreflective Cartesian closed subcategory of $$\mathbf{Top}$$ or $$\mathbf{Haus}$$ containing $$C(a)\times C(b)$$ if $$a\neq b$$.

This is a specific example showing that there is no largest coreflective Cartesian subcategory of $$\mathbf{Top}$$ or $$\mathbf{Haus}$$.