nLab
subsequential space

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 XX equipped with a relation between sequences and points, called “converges to,” with the following properties.

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

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

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

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

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 SeqTopSeqTop of sequential (topological) spaces is a full reflective subcategory of the category SeqPsTopSeqPsTop 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 SeqTopSeqTop is coreflective in TopTop.

Furthermore, SeqPsTopSeqPsTop 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 SeqTopSeqTop in SeqPsTopSeqPsTop 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 TopTop and then regard the result as a subsequential space, or carry out the construction in SeqPsTopSeqPsTop to begin with.

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