© 1996 Copyright by Oren Kolman. All rights reserved.

Question

Quasi-minimal spaces in singular and successor cardinalities Oren Kolman

Notation: Let k be an infinite cardinal, T(k) the family of topological spaces of cardinality k, and T(k,2) the Hausdorff members of T(k).

Definitions:

1. A space X is weakly quasi minimal (wqm) iff every subspace Y of cardinality |X| contains a homeomorphic copy of X.

2. A family F supports a family G iff every member of G contains a homeomorphic copy of a member of F.

3. Qmin(k) means:

   1. neither T(k) nor T(k,2) is supported by its wqm members, and

   2. any supporting family of T(k) or T(k,2) has more than k members.

1. If k = \aleph_\omega is less than the continuum, is Qmin(k) true, false, or independent of ZFC?

2. Does ZFC imply Qmin(k+)?

3. If k > ck(k) = \aleph₀ and k is not a strong limit, is Qmin(k) true, false, or independent of ZFC? (the general version of Q1)

In 1994, Matthews and McMaster [1] defined and studied Qmin(k), proving:

1. GCH implies that Qmin(k+) holds for every infinite cardinal k;

2. if the continuum c is a regular cardinal, then Qmin(c) is true.

The Hajnal-Juhasz theorem implies that Qmin(k) is false if k is a singular strong limit cardinal. If k is weakly compact, Qmin(k) is false too. Further, ZFC suffices to prove that Qmin(c) holds. Taking these facts together, one obtains:

1. Con(ZFC) implies Con(ZFC + for all k Qmin(k) is true iff k is an infinite successor cardinal);

2. Qmin(k) is independent of ZFC if k is a singular cardinal of uncountable cofinality;

3. if there is a weakly inaccessible cardinal, then Con(ZFC+ Qmin(k) is true for some weakly inaccessible cardinal k);

4. if k has the tree property (no k-Aronszajn trees), then Con(ZFC + GCH + Qmin(k) is false for some (weakly compact) inaccessible cardinal k).

The tree property for k does not imply that Qmin(k) is false (using a model of W. Mitchell).

In light of these results, questions (1), (2) and (3) are quite natural.

Using large families of almost disjoint sets (which exist under hypotheses weaker than GCH), it has been shown that:

1. for every infinite k, Qmin(k+) holds;

2. Qmin(k+) does not imply 2^k = k+.

References

1. P. Matthews and T.B.M.McMaster, A viewpoint on minimality in topology, Irish Math. Soc. Bull., 32 (1994), 7-16.

2. E. Coleman (O. Kolman), Toronto spaces, minimality, and a theorem of Sierpinski, Irish Math. Soc. Bull., 38 (1997), 53-65.

3. O. Kolman, Independence results on quasi-minimal Hausdorff spaces in successor cardinalities. (Submitted)