Irresolvable and submaximal spaces: Homogeneity versus σ-discreteness and new ZFC examples
✍ Scribed by O.T. Alas; M. Sanchis; M.G. Tkac̆enko; V.V. Tkachuk; R.G. Wilson
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 137 KB
- Volume
- 107
- Category
- Article
- ISSN
- 0166-8641
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✦ Synopsis
An example of an irresolvable dense subspace of {0, 1} c is constructed in ZFC. We prove that there can be no dense maximal subspace in a product of first countable spaces, while under Booth's Lemma there exists a dense submaximal subspace in [0, 1] c . It is established that under the axiom of constructibility any submaximal Hausdorff space is σ -discrete. Hence it is consistent that there are no submaximal normal connected spaces. If there exists a measurable cardinal, then there are models of ZFC with non-σ -discrete maximal spaces. We prove that any homogeneous irresolvable space of non-measurable cardinality is of first category. In particular, any homogeneous submaximal space is strongly σ -discrete if there are no measurable cardinals.