Atlas: Strong irresolvability vs. irresolvability by W. W. Comfort

Atlas home || Conferences | Abstracts | about Atlas

Spring Topology and Dynamical Systems Conference 2003
March 20-22, 2003
Texas Tech University
Lubbock, TX, USA

Organizers
Wayne Lewis, Razvan Gelca, Harold Bennett, Carl Seaquist

Strong irresolvability vs. irresolvability
by
W. W. Comfort
Wesleyan University
Coauthors: Wanjun Hu

Responding to a question of Eckertson [Topology Appl. 79 (1997), 81-11], the present authors showed [same J. 127 (2003), 343-354] in ZFC that for k ³ w some family A Í P(k) with |A|=2^k is simultaneously maximal w-independent and (maximal) k-independent. This yields a space X=(X, T)=È_{h < k} D_h Í K:={0, 1}^{2^k} such that (a) each D_h is dense in K; (b) the sets D_h are pairwise homeomorphic; (c) each nonempty (relatively) open U Í D_h satisfies |U|=d(U)=k; and (d) no nonempty subset of D_h admits complementary dense subsets. The method of KID-expansions is applied to the space (X, T) Í K to produce in the case k = w a Tychonoff topology U Ê T such that the countable space (X, U) is extraresolvable in the sense of Malykhin, but not strongly extraresolvable.

Remarks. (1) Comfort and García-Ferreira [Topology Proc. 23 (1998), 45-74] had showed the existence of extraresolvable but not strongly extraresolvable Tychonoff spaces of arbitrary cardinality k > w, and had raised the question concerning the case k = w.
(2) García-Ferreira and González-Silva [Topology Appl. 122 (2002), 151-156] have achieved a quite different example answering the same question.
(3) The question of García-Ferreira, Malykhin and Tomita [same J. 101 (2000), 257-271], whether there exists an uncountable extraresolvable space which is not maximally resolvable, remains open in ZFC.

Date received: February 24, 2003