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 $\left(x\right)$ converges to $x$.

2. If a sequence $\left({x}_{n}\right)$ converges to $x$, then so does any subsequence of $x$.

3. If, for some sequence $\left({x}_{n}\right)$ and some point $x$, every subsequence of $\left({x}_{n}\right)$ contains a further subsequence converging to $x$, then $\left({x}_{n}\right)$ itself converges to $x$.

The final property can be stated less constructively as “if $\left({x}_{n}\right)$ does not converge to $x$, then there is a subsequence $\left({x}_{{n}_{k}}\right)$ of $\left({x}_{n}\right)$ such that no subsequence of $\left({x}_{{n}_{k}}\right)$ 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.

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 $\mathrm{SeqTop}$ of sequential (topological) spaces is a full reflective subcategory of the category $\mathrm{SeqPsTop}$ 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 $\mathrm{SeqTop}$ is coreflective in $\mathrm{Top}$.

Furthermore, $\mathrm{SeqPsTop}$ 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 $\mathrm{SeqTop}$ in $\mathrm{SeqPsTop}$ 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 $\mathrm{Top}$ and then regard the result as a subsequential space, or carry out the construction in $\mathrm{SeqPsTop}$ to begin with.

It follows that the geometric realization functor from simplicial sets can equally well be regarded as landing in $\mathrm{Top}$, $\mathrm{SeqTop}$, or $\mathrm{SeqPsTop}$. Of course, it has a singular complex functor as a right adjoint in any of these three cases. In the cases of $\mathrm{SeqTop}$ and $\mathrm{SeqPsTop}$, geometric realization actually preserves all finite limits; in fact it and the singular complex functor form a geometric morphism between $\mathrm{SimpSet}$ and a Grothendieck topos that contains $\mathrm{SeqPsTop}$ as a reflective subcategory (the “topological topos” of Johnstone’s paper). Recall that geometric realization landing in $\mathrm{Top}$ doesn’t even preserve finite products, unless we replace $\mathrm{Top}$ 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 $\mathrm{Conv}$ of convergence spaces is also a complete and cocomplete quasitopos (hence, in particular, locally cartesian closed) and includes all of $\mathrm{Top}$ as a reflective subcategory. However, $\mathrm{Conv}$ is not locally presentable and has no generator, and while the embedding of $\mathrm{Top}$ into $\mathrm{Conv}$ also preserves all limits (since it has a left adjoint), it actually preserves fewer colimits than the embedding of $\mathrm{SeqTop}$ into $\mathrm{SeqPsTop}$. 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 $\mathrm{Conv}$, in general the result won’t even be a topological space.

References

• P. T. Johnstone, On a topological topos. Proc. London Math. Soc. (3) 38 (1979) 237–271

Revised on May 24, 2010 16:01:38 by Todd Trimble (69.118.56.215)