On Weak Reflections in Some Classes of Topological Spaces
by
Martin Maria Kovár
Technical University of Brno

The topic of weak reflections namely in compact spaces a has nice and long history. First I heard the beginning of the story in 1990 by J. Rosický and then by M. Husek during my studies. I heard that it was perhaps Z. Frolík who 35 years ago in some occasion mentioned the question: Is there a compactification cX of a topological space X such that every continuous mapping from X into any compact space Y can be continuously extended to cX? In other words: Is the class of compact spaces weakly reflective in the class of topological spaces? It is probably impossible to find who first asked the question and also it is difficult to mention all people who worked in the related topics. I know that it was - for example, in the alphabetic order - J. Adámek, A. Dow, H. Herrlich, M. Husek, J. Rosický, S. Salbany, S. Todorcev and S. Watson but perhaps also some other colleagues e.g. from younger generation of mathematicians. Now, the question is stated explicitly in [AR] and [He1]. S. Todorcevi\'c posed a modified problem [DW]: Does there exist a space U such that every topological space X has a compactification cX embeddable into a power of U such that every continuous mapping from X into a compact T1 space has a continuous extension onto cX?

In a discussion in Brno in 1990 J. Rosický mentioned Z. Frolík's question to me as an interesting unsolved problem. But already in 1991 M. Husek answered Frolík's question and also Todorcevi\'c's problem in the negative. In fact, Husek's main results were the following:

Let X be a topological space. Then:

1. If the Wallman remainder of X is finite, then the Wallman compactification of X is the weak reflection of X in compact spaces.
2. X contains an infinite family {X(n)} of closed noncompact subsets such that the intersection of X(n), X(m) is compact for different n, m then X has no weak reflection in compact spaces.
3. A normal T1 space has a weak reflection in compact spaces iff its \'Cech-Stone remainder is finite.

Regarding the weak reflections in compact spaces now all seemed to be done. When reading Husek's article in 1994 again, I tried to construct few simple examples of spaces with or without weak reflections in compact spaces. But all my examples satisfied either the condition (1) or the condition (2). In other words, I was not able to find a space with the infinite Wallman remainder which did not contain the family {X(n)}. M. Husek in [Hu] (in a joint discussion with colleagues from Univ. of Kansas) derived that if the Wallman remainder of X contains an infinite discrete subspace, then X contain the family {X(n)} and hence has no weak reflection in compact spaces. In [Ko2] I proved a similar result regarding the case that the Wallman remainder of X contains an infinite subspace with the co-finite topology. But the Wallman remainder of any space always is T1 and it is not very difficult to show that every infinite T1 space must contain an infinite subspace, which has either the discrete or the co-finite topology. Hence I was able to improve Husek's results (1) - (3) to the following:

4. A space has a weak reflection in compact spaces iff its Wallman remainder is finite.

Now the the spaces having a weak reflection in compact spaces are fully characterized. Nevertheless, new unsolved problems arise when one replace compact spaces by another classes. M. Husek proved that the class of compact spaces is not weakly reflective in topological spaces but it is natural to study the weak reflections "below" as well as "above" compactness. In my talk I will speak about the weak reflections in classes that contain the class of compact spaces and that are contained in the class of \theta-regular spaces (which are described in detail in [Ja], [Ko1] and [Ko3]). In other words, the studied classes will be characterized by properties bounded above by compactness and below by \theta-regularity. I will present some characterization and comparative theorems with focus in the following problems:

5. Characterize the spaces which have a weak reflection in locally compact spaces.
6. Exists there a weakly reflective subclass of topological spaces that contains the class of compact spaces and is contained in \theta-regular spaces?

It should be noted that by a locally compact space we mean a space in which each point has a closed compact neighborhood. The class of such defined locally compact spaces obviously is not weakly reflective, but it is interesting to know whether the spaces having a weak reflection in this class have a characterization similar to (4).

References

[AR] Adámek J., Rosický J., On injectivity in locally presentable categories, Trans. Amer. Math. Soc. 336, 2, (1993), 785-804.

[DW] Dow A., Watson S., Universal spaces, preprint (June 1989).

[He1] Herrlich H., Almost reflective subcategories of Top, preprint (1991) (to appear in in Topology and its Appl.).

[He2] Herrlich H., Compact T0-Spaces and T0-Compactifications, Applied Categorial Structures 1 (1993), 111-132.

[Hu] Husek M., \'Cech-Stone-like compactifications for general topological spaces, Comment. Math. Univ. Carolinae 33, 1 (1992), 159-163.

[Ja] Jankovi\'c D. S., \theta-regular spaces, Internat. J. Math. Sci. 8 (1986), no. 3, 615-619.

[Ko1] Kovár M. M., On \theta-regular spaces, Internat. J. Math. Sci. 17 (1994), no. 4, 687-692.

[Ko2] Kovár M. M. Which topological spaces have a weak reflection in compact spaces?, Comment. Math. Univ. Carolinae 36, 3 (1995), 529-536.

[Ko3] Kovár M. M. A remark on \theta-regular spaces Internat. J. Math. Sci. 21 (1998), no.1, 199-200

Date received: June 26, 2000