non-symmetric version of compact = totally bounded + complete

Question

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is there a similar such characterization of compactness for quasi uniform spaces?

Answer 1

This answer is preliminary. Now I am looking in the papers and soon I shall extend the answer.

I am related with the subjects, but I am not a specialist. I looked into papers and can say you the following. If you are so interested and want to have a perfect answer, you may ask Hans-Peter Künzi (use this e-mail: Hans-peter.Kunzi (at) uct.ac.za).

So, IMHO, it seems the following.

A quasi-uniformity on a set $X$ is a filter $\mathcal U$ on $X\times X$ satisfying the conditions: Each $U\in\mathcal U$ is reflexive, and for each $U\in\mathcal U$ there is $V\in\mathcal U$ such that $V^2:=V\circ V\subset U$. Here $$V\circ V :=\{(x, z)\in X\times X:\mbox{ there is }y\in X\mbox{ such that }(x, y)\in V\mbox{ and }(y, z)\in V\}.$$ For each quasi-uniformity $\mathcal U$ the filter $\mathcal U^{-1}$ consisting of the inverse relations $$U^{-1}=\{(y, x)\in X\times X: (x, y)\in U\}$$ where $U\in\mathcal U$ is called the conjugate quasi-uniformity of $\mathcal U$. A quasi-uniformity $\mathcal U$ is said to be a uniformity if $\mathcal U^{-1}=\mathcal U.$ If $\mathcal U_1$ and $\mathcal U_2$ are quasi-uniformities on $X$ such that $\mathcal U_1\subset \mathcal U_2$, then $\mathcal U_1$ is called coarser than $\mathcal U_2$. The coarsest uniformity $\mathcal U^s$ finer than $\mathcal U$ is generated by the subbase $\mathcal U\cup\mathcal U^{-1}$. [Kü1]

A quasi-uniform space $(X,\mathcal U)$ is called precompact (resp. totally bounded) if for each $U\in\mathcal U$, a cover $\{U(x): x\in X\}$ has a finite subcover (resp. $\mathcal U^s$ is precompact). A uniform space is totally bounded iff it is precompact. Every totally bounded quasi-uniform space is precompact but the converse does not hold, in general. [Lam] A quasi-uniform space $(X,\mathcal U)$ is totally bounded iff it is doubly hereditarily precompact, that is both $\mathcal U$ and $\mathcal U^{-1}$ are hereditarily precompact. [Kü1] Each compact quasi-uniform space is precompact, but the converse does not hold, in general, too. As a simple example I can propose Sorgenfrey circle that is the unit circle $\Bbb T=\{e^{i\varphi}:\varphi\in\Bbb R\}$ endowed with a quasi-uniformity generated by a base $\{U_n\}$, where $U_n=\{(x,y)\in\Bbb T: y=xe^{i\varphi}$ and $\varphi\in [0,1/n)\}.$ Sorgenfrey circle is a precompact quasi-uniform space, which is neither compact nor totally bounded. On the other hand, a quasi-uniform space is precompact if and only if every ultrafilter is a Cauchy filter [FL]. At last, there exists a compact quasi-uniform space which is not totally bounded: let $a<b$ be real numbers; endow the set of the reals with a quasi-uniformity generated by a base $\{\{(x, y):x = y$ or $a < x < b\}\}$ [Lam]. From the other side, for an arbitrary topological space $X$ the Pervin quasi-uniformity is the finest compatible totally bounded quasi-uniformity of $X$. I recall that each topological space $(X; \tau)$ is quasi-uniformizable, since $\tau$ coincides with the topology induced by the Pervin quasi-uniformity generated by the subbase $\{[X\setminus G\times X]\cup [X \times G] : G\in\tau\}$. [Kü1]

There are several notions of completeness for quasi-uniform spaces [Kü1-2].

A quasi-uniform space $(X, \mathcal U)$ is called bicomplete provided that $\mathcal U^s$ is a complete uniformity. So, a quasi-uniform space $(X, \mathcal U)$ is totally bounded and bicomplete iff $(X, \mathcal U^s)$ is a complete precompact uniform space iff $(X, \mathcal U^s)$ is compact. In this case the space $(X, \mathcal U)$ is compact too. Sorgenfrey circle is bicomplete and precompact, but it is neither compact nor totally bounded.

A topological space is supersober if the convergence set of each convergent ultrafilter is the closure of some unique singleton. In particular, each Hausdorff space is supersober. The class of locally compact strongly sober spaces (sometimes also called skew compact; see Kopperman [Ko, Comment 4.12]), is a suitable analogue in the category $Top_0$ of $T_0$-spaces to the class of compact spaces in the category of Hausdorff spaces. Here “strongly sober” means “compact and supersober”. The strongly sober locally compact spaces (see also the articles of Hötzel Escardó [HE], Lawson [Law], Salbany and Todorov [ST] and Smyth [Sm] for further details) can be characterized as the topological $T_0$-spaces that admit a totally bounded bicomplete quasi-uniformity. It is the coarsest compatible quasi-uniformity of such spaces. [Kü2]

A filter $\mathcal F$ on a quasi-uniform space $(X, \mathcal U)$ is said to be left $K$-Cauchy (resp. right $K$-Cauchy) if for each $U\in \mathcal U$ there is $F\in \mathcal F$ such that $U(x)\in \mathcal F$ (resp. $U^{-1}(x)\in \mathcal F$) whenever $x\in F$. A quasi-uniformity is called left (resp. right) $K$-complete provided that each left (resp. right) $K$-Cauchy filter converges. A quasi-uniform space is compact iff it is precompact and left $K$-complete [Kü3]. A quasi-uniform space $(X, \mathcal U)$ is hereditarily precompact if and only if each ultrafilter is left $K$-Cauchy (equivalently, each filter is co-stable). Here a filter is called co-stable if it is stable in the conjugate space. A filter $\mathcal F$ on a quasi-uniform space $(X, \mathcal U)$ is called stable if $\bigcap_{F\in\mathcal F} U(F)\in\mathcal F$ whenever $U\in\mathcal U$. [Kü1]

References

[FL] P. Fletcher, W.F. Lindgren, Quasi-Uniform Spaces, Lecture Notes Pure Appl. Math. Vol. 77, Dekker, New York, 1982.

[HE] M. Hötzel Escardó , The regular-locally compact core of a stably locally compact locale, J. Pure Appl. Algebra, 157, 41-55.

[KMRV] H-P. A. Künzi, M. Mršević, I.L. Reilly, M.K. Vamanamurthy, Convergence, precompactness and symmetry in quasi-uniform spaces, Mathematica Japonica, 01/1993, 38:2.

[Ko] R. D. Kopperman, Asymmetry and duality in topology, Topology Appl., 66, 1-39.

[Kü1] H-P. A. Künzi, Quasi-uniform spaces, (April 28, 2001).

[Kü2] H-P.A. Künzi, Quasi-uniform spaces in the year 2001, (? in Recent Progress in General Topology, II, 313-344, North-Holland, Amsterdam, 2002).

[Kü3] H-P.A. Künzi, Nonsymmetric topology, in Topology with Applications, Bolyai Soc. Math. Studies, Vol. 4, Szekszárd, 1993, 303-338.

[Kü4] H-P.A. Künzi, Totally bounded quiet quasi-uniformities, Topology Proceedings, 01/1990, 15.

[Lam] P. Th. Lambrinos, On precompact (quasi-) uniform structures, Proc. Amer. Math. Soc., 62:2, 1977, 365-366.

[Law] J. D. Lawson, Order and strongly sober compactifications, in: Topology and Category Theory in Computer Science, Reed, G. M., A. W. Roscoe and R. F. Wachter, eds., Clarendon Press, Oxford, 179-205.

[Sm] M. B. Smyth, Stable compactification I, J. London Math. Soc., 45, 321-340.

[ST] S. Salbany, T. Todorov. Nonstandard analysis in topology: Nonstandard and standard compactifications, Journal Symbol. Logic, 65, 1836-1840.