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?

]]>Yes, fixed.

]]>I presume you meant “can *not* be proved to be sequential”.

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.

]]>Interestingly ELS are often thought of as theoretical computer scientists, perhaps more than categorical topologists.

]]>@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.

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 :)).

]]>I don’t suppose that sequential spaces are an exponential ideal in subsequential spaces?

]]>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.

]]>This surprises me. Without Hausdorff criterium ? What is the inner hom ? Usual $C(X,Z)$ in compact-open topology ?

]]>Added to sequential topological space the observation that the category of such is cartesian closed.

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