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.
A subsequential space is a set equipped with a relation between sequences and points, called “converges to,” with the following properties.
For every , the constant sequence converges to .
If a sequence converges to , then so does any subsequence of .
If, for some sequence and some point , every subsequence of contains a further subsequence converging to , then itself converges to .
The final property can be stated less constructively as “if does not converge to , then there is a subsequence of such that no subsequence of converges to .”
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.
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 of sequential (topological) spaces is a full reflective subcategory of the category 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 is coreflective in .
Furthermore, 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 in 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 and then regard the result as a subsequential space, or carry out the construction in to begin with.
It follows that the geometric realization functor from simplicial sets can equally well be regarded as landing in , , or . Of course, it has a singular complex functor as a right adjoint in any of these three cases. In the cases of and , geometric realization actually preserves all finite limits; in fact it and the singular complex functor form a geometric morphism between and a Grothendieck topos that contains as a reflective subcategory (the “topological topos” of Johnstone’s paper). Recall that geometric realization landing in doesn’t even preserve finite products, unless we replace 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 of convergence spaces is also a complete and cocomplete quasitopos (hence, in particular, locally cartesian closed) and includes all of as a reflective subcategory. However, is not locally presentable and has no generator, and while the embedding of into also preserves all limits (since it has a left adjoint), it actually preserves fewer colimits than the embedding of into . 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 , in general the result won’t even be a topological space.