© 1996, Topology Atlas
The product of any family of Tychonoff spaces is Tychonoff. Furthermore all lower separation properties are preserved under the product operation. Normality, however, is very poorly behaved in this respect, even when one of the factors is very nice. For example, there is a normal space whose product with the irrationals, a separable metric space, is not normal. Even when one of the factors is compact normality may not be preserved: omega_1 endowed with the order topology is a normal space, omega_1+1 is compact, but omega_1 x omega_1+1 is not normal (for more on these and related spaces see the survey [P]). Whether normality is preserved when one of the factors is a compact metrizable space is a much more difficult question.
In 1951, C.H. Dowker characterized those spaces whose product with the closed unit interval [0,1] is normal. He proved
Theorem ([D]) The following are equivalent for a space X:
A space is defined to be countably paracompact if every countable open cover has a locally finite refinement. In the same paper Dowker asked for a counterexample. Any normal space X for which X times [0,1] is not normal has therefore since been known as a Dowker space.
Dowker spaces are notoriously difficult to construct. It took over twenty years for the first ZFC example to be constructed (by M.E. Rudin in 1972 [R1]) and another twenty plus years for the second ZFC example (in 1994 due to Z. Balogh [B]). Before and in-between an incredible variety of other Dowker spaces were constructed from set-theoretic axioms beyond the usual axioms of ZFC.
Rudin's 1972 example was quite large: of cardinality and weight aleph_omega^omega. So instead of closing the book on Dowker spaces her example fueled the search for so called 'small' Dowker spaces, one with some countable local or global property. Consistently there are such examples. One may construct first countable Dowker spaces of size omega_1 from CH and weaker axioms. Likewise more esoteric examples can be constructed from stronger assumptions (see the surveys [R2] and [SW]). Almost any property not possessed by Rudin's space generated an interesting and difficult open problem.
Balogh's example is the first small Dowker space in ZFC. It is of size and weight continuum and in addition it answered many other questions: it is hereditarily normal and sigma-discrete (Rudin's space satisfied neither). However, important open problems concerning small Dowker spaces remain:
Question Is there a first countable Dowker space? Or one of size omega_1? Or one with both of these properties?
Recently M. Kojman and S. Shelah constructed in ZFC a Dowker subspace of Rudin's space of size aleph_omega+1. It may be consistent that this is the smallest Dowker space.
Concerning the existence of Dowker spaces with more esoteric properties, the most interesting and most difficult open problem is
Question Is there a Dowker space with a sigma-disjoint base?
Even a consistent example of such a space is not known.
References
[B] Z. Balogh, A small Dowker space in ZFC, Proc. Amer. Math. Soc. to appear.
[Do] C.H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951) 219-224.
[K] M. Kojman and S. Shelah, A ZFC Dowker space in aleph_omega+1: an application of pcf theory to topology, preprint.
[R1] M.E. Rudin, A normal space X for which X x I is not normal, Fund. Math. 73 (1972) 179-186.
[R2] M.E. Rudin, Dowker spaces, in: Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, Eds., (North-Holland, Amsterdam-New York-Oxford, 1984), 761-780.
[SW] P.J. Szeptycki and W.A.R. Weiss, Dowker Spaces, in: The Work of Mary Ellen Rudin, ed. Frank Tall, The New York Acadamy of Sciences 705 (1993) 119-129.