© 1996, Topology Atlas

A (Tychonoff) space which is homeomorphic to a closed subset of a product of real lines is said to be realcompact. There are many known characterizations of realcompactness among them some involving z-ultrafilters on the space, the Stone-Cech compactification of the space, or the ring of continuous functions on the space. Not every space is realcompact. For example, every realcompact pseudocompact space is compact so an ordinal of uncountable cofinality is not realcompact. On the other hand, there are large classes of spaces that are realcompact. Every metric space of non-measurable cardinality is realcompact, and every Lindelof space is realcompact. Since every space is the continuous image of a discrete space of the cardinality of the space, realcompactness is not preserved by continuous maps; omega_1 is a non-realcompact space which is the continuous image of the discrete space of cardinality omega_1, which is realcompact. On the other hand, Lindelofness is preserved by continuous maps, so every continuous image of a Lindelof space is realcompact. Mrowka (E-complete regularity and E-compactness, in General Topology and Its Relations to Modern Analysis and Algebra, (Academic Press, New York, 1971) 207-214), and later Arhangel'skii and Okunev (Characterization of properties of spaces by properties of their continuous images, Vestnik Moskov. Univ. 40 (1985) 28-30), asked about the converse of this observation: If every continuous image of a space is realcompact, must the space be Lindelof?

This question remains open, but some progress has been made. Among the results proved in (Eckertson, Fleissner, Korovin, Levy, Not realcompact images of not Lindelof spaces, Top. and Its App. 58 (1994) 115-125) are the following: If a space maps continuously onto its square, and if every continuous image of the space is realcompact, then the space (and, therefore, its square) is Lindelof. If every continuous image of a space is realcompact, then every closed, discrete subset of the space is Lindelof. (The conclusion in this statement is phrased to make the result sound closer to an answer to the Mrowka-Arhangel'skii-Okunev question than it really is; the point is that any example providing a negative answer must have countable extent.) If a space has Lindelof degree at most the cardinal of the continuum and if every continuous image of the space is realcompact, then the space is Lindelof. Thus, if there is a non-Lindelof space each of whose continuous images is realcompact, it must be small in the sense that all closed discrete subsets are small, but it must be large in the sense that the Lindelof degree is large.

The result that there is no example with large Lindelof degree raises a reflection question which may be of interest in its own right. Suppose a space X embeds as a closed subset of a product of real lines in such a way that every projection onto continuum many coordinates is Lindelof. Is X necessarily Lindelof? (The assumption that the subset be closed is easily seen to be essential.) An affirmative answer to this question implies an affirmative answer to the Mrowka-Arhangel'skii-Okunev question.