Subsequential spaces

Idea

A subsequential space is a set equipped with a notion of sequential convergence, giving it a “topology” in an informal sense.

Any topological space (or more generally, any pseudotopological space) becomes a subsequential space with its standard notion of convergence, but only for a sequential space can the topology be recovered from sequential convergence. In the other direction, not every subsequential space is induced by a topological one. Despite these apparent drawbacks, subsequential spaces have a number of advantages; see below.

Definition

A subsequential space is a set $X$ equipped with a relation between sequences and points, called “converges to”, with the following properties:

1. For every $x\in X$, the constant sequence $(x)$ converges to $x$.

2. If a sequence $(x_n)$ converges to $x$, then so does any subsequence of $x$.

3. If, for some sequence $(x_n)$ and some point $x$, every subsequence of $(x_n)$ contains a further subsequence converging to $x$, then $(x_n)$ itself converges to $x$.

The final property can be stated less constructively as “if $(x_n)$ does not converge to $x$, then there is a subsequence $(x_{n_k})$ of $(x_n)$ such that no subsequence of $(x_{n_k})$ converges to $x$.”

Note that this definition matches the definition of pseudotopological space except for the restriction to sequences instead of general nets. Accordingly, one may call a subsequential space a sequential pseudotopological space.

Subsequential spaces are also known as Kuratowski limit spaces, or L-spaces; see Menni & Simpson (2002).

A subsequential space is said to be sequentially Hausdorff if each sequence converges to at most one limit.

Properties

The definition of a subsequential space is arguably easier and more intuitive than that of a topological space. Continuity of functions between subsequential spaces is likewise easy to define by preservation of convergent sequences.

As mentioned above, the category $SeqTop$ of sequential (topological) spaces is a full reflective subcategory of the category $SeqPsTop$ of subsequential spaces. Thus, subsequential spaces include many spaces of interest to topologists, including all metrizable spaces and all CW complexes, and so they can be regarded as a sort of nice topological space.

Not every subsequential space is a sequential (topological) space, but somewhat surprisingly, every sequentially Hausdorff subsequential space is necessarily a sequential space. Note, though, that while any Hausdorff space is sequentially Hausdorff, the converse is not true even for sequential spaces (though it is true for first-countable spaces). Also of note is that $SeqTop$ is coreflective in $Top$.

Furthermore, $SeqPsTop$ is also a nice category of spaces: it is locally cartesian closed and in fact a quasitopos. Since it is a “Grothendieck quasitopos” (the category of presheaves on a category which are sheaves for one Grothendieck topology and separated for another one), it is also locally presentable. In particular, it is complete and cocomplete, and has a small generating set.

Of course, the embedding of $SeqTop$ in $SeqPsTop$ preserves all limits, since it has a left adjoint, but somewhat surprisingly it also preserves many colimits. In particular, it preserves all the colimits used in the construction of a CW complex; thus it makes no difference whether you carry out the construction of a CW complex in Top and then regard the result as a subsequential space, or carry out the construction in $SeqPsTop$ to begin with.

It follows that the geometric realization functor from simplicial sets can equally well be regarded as landing in $Top$, $SeqTop$, or $SeqPsTop$. Of course, it has a singular simplicial complex functor as a right adjoint in any of these three cases (by nerve and realization). In the cases of $SeqTop$ and $SeqPsTop$, geometric realization actually preserves all finite limits; in fact it and the singular complex functor form a geometric morphism between $SimpSet$ and a Grothendieck topos that contains $SeqPsTop$ as a reflective subcategory (the “topological topos” of Johnstone’s paper). Recall that geometric realization landing in $Top$ doesn’t even preserve finite products, unless we replace $Top$ by (for instance) compactly generated spaces.

These properties of subsequential spaces should be compared with analogous ones for convergence spaces and their relatives, such as pseudotopological spaces. The category $Conv$ of convergence spaces is also a complete and cocomplete quasitopos (hence, in particular, locally cartesian closed) and includes all of $Top$ as a reflective subcategory. However, $Conv$ is not locally presentable and has no generator, and while the embedding of $Top$ into $Conv$ also preserves all limits (since it has a left adjoint), it actually preserves fewer colimits than the embedding of $SeqTop$ into $SeqPsTop$. In particular, it does not preserve the colimits used in the construction of CW complexes: if you carry out the construction of a CW complex in $Conv$, in general the result won’t even be a topological space.

 See also

• sequential preconvergence space

References

• Peter Johnstone, On a topological topos. Proc. London Math. Soc. (3) 38 (1979) 237–271 doi:10.1112/plms/s3-38.2.237

• Matias Menni, Alex Simpson: Topological and Limit-space Subcategories of Countably-based Equilogical Spaces, Math. Struct. in Comp. Science 12 (2002) 739-770 [pdf, doi:10.1017/S0960129502003699]

• Sean Moss, Blog post at the $n$-Category café