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Added to sequential topological space the observation that the category of such is cartesian closed.
This surprises me. Without Hausdorff criterium ? What is the inner hom ? Usual $C(X,Z)$ in compact-open topology ?
Zoran, I only recently learned of this result myself. A reference for this fact is given in the references at convenient category of topological spaces: see the paper by Escardo, Lawson, and (Alex) Simpson, all acknowledged researchers in categorical topology. They are not the first to prove the result, I understand, but they did set out a general context for which this result is a special case.
I don’t think the Hausdorff condition is required. My understanding is that one basically takes the compact-open topology (this might have to be adjusted in the non-Hausdorff case though; I’ll explain in a moment) on the function space, and then adjusts this function space topology by coreflecting back from $Top$ to sequential spaces (i.e., applying the right adjoint to the inclusion), just as one does in the compactly generated case.
(As I say, there may be extra fiddlyness in the non-Hausdorff case, but here the correct function space topology is described in the article on exponential laws for spaces, under the subsection on core-compact spaces, which precisely characterize the exponentiable objects in $Top$, here.)
The general context of ELS is that they consider a class of $Top$-exponentiable spaces $\mathcal{C}$ with the property that the product of any two spaces in $\mathcal{C}$ is a $Top$-colimit of spaces in $\mathcal{C}$. The theorem is that the full subcategory of $Top$ whose objects are $Top$-colimits of spaces in $\mathcal{C}$ is then cartesian closed. This applies in particular if we take $\mathcal{C}$ to consist of just the one-point compactification of $\mathbb{N}$ with the discrete topology; here the resulting cartesian closed category is that of sequential spaces.
I don’t suppose that sequential spaces are an exponential ideal in subsequential spaces?
Todd, you are bringing beautiful stuff (I thought recently after going into much detail about nonHausdorff compactly generated spaces that I understand somewhat in detail the exponential law, but now this beautiful general theorem is a whole vast area :)).
@Mike: well, I don’t know! But it doesn’t seem at all implausible (and it would be interesting and possibly original if true): as you know, sequential spaces are a reflective subcategory of subsequential spaces, and subsequential spaces form a quasitopos (which of course is cartesian closed), so all we would need is that the reflection preserves products, right? Hm…
@Zoran: what’s also intriguing is how soft the relevant proofs of ELS are, which makes me wonder about the true generality of the result. Could something like this be true for much more general categories besides $Top$? I don’t know, but this looks interesting.
Interestingly ELS are often thought of as theoretical computer scientists, perhaps more than categorical topologists.
I wrote about why metric spaces can’t be proved to be sequential (not even the real line) in constructive mathematics.
Actually, all that I really showed is how the usual proof uses excluded middle (and also countable choice). It would be nice to find, say, a specific topos in which the real line is not sequential. (It would be even better to find two examples: one in which excluded middle holds but countable choice fails, and one in which countable choice holds but excluded middle fails.)
One might also hope for a constructive version of sequentiality: something classically equivalent to the usual definition, while constructively holding of all first-countable spaces (or at least metrizable spaces). But I doubt it; constructive analysis usually just bites the bullet and uses nets.
I presume you meant “can not be proved to be sequential”.
Yes, fixed.
The first topos that comes to my mind to look at is the topos of sheaves on the space of real numbers, in which $R_d$ is the sheaf of continuous real-valued functions. What does it mean for a sub-sheaf of that to be open or sequentially open?
Added the property
- The category of sequential spaces embeds fully faithfully into the category of light condensed sets.
This needs a proof or a reference.
I’m planning to put in some more content on light condensed sets at the appropriate page, as I get the time. I will come back and put in the reference to Scholze’s lecture on this.
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