nLab quasi-topological space

Idea

The category of quasi-topological spaces was proposed in Spanier 1963 as a substitute for the category Top of ordinary topological spaces, in order to serve as a convenient category for the purposes of algebraic topology. In particular, quasi-topological spaces form a complete and cocomplete cartesian closed category.

Today, quasi-topological spaces seem to be regarded mostly as a historical curiosity, perhaps because working topologists were never comfortable with the set-theoretic issues that accompany them. In retrospect, however, they are an impressive testament to the conceptual insight of Spanier into ideas of topos theory which were at the time (early 1960’s) barely in the air, and even not quite born yet (being an early example of quasitopos, whose name perhaps derives from Spanier’s notion, compare Dubuc & Español 2006, p. 12).

Definition

Let $\mathcal{C H}$ be the category of compact Hausdorff spaces. This may be regarded as a (large) site (in fact a concrete site) when equipped with the Grothendieck pretopology of finite families $\{ C_\alpha \to X \}_{ \alpha \in I}$ for which the induced map $\coprod_{\alpha\in I} C_\alpha \to X$ is surjective.

Definition

A quasi-topological space is a (small-set valued) concrete sheaf on $\mathcal{C H}$.

The (super-large) category of quasi-topological spaces is a quasitopos (although this is not immediately obvious for size reasons — in particular, it is probably not a Grothendieck quasitopos). In particular, it is a locally cartesian closed category.

References

• Edwin Spanier, Quasi-topologies, Duke Mathematical Journal 30, number 1 (1963) pp 1–14 (doi:10.1215/S0012-7094-63-03001-1)

• Eduardo Dubuc, Luis Español, Quasitopoi over a base category (arXiv:math.CT/0612727)