Ofelia T. Alas

Irresolvable and submaximal spaces: Homogeneity versus σ-discreteness and new ZFC examples1

Abstract 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 Booths 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.

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω1-monolithic compact space X, if Cp(X)is star countable then it is Lindelöf.

Connectifying some spaces

Abstract A Hausdorff space X is called (countably) connectifiable if there exists a connected Hausdorff space Y (with |Y⊮X| ⩽ ω ; respectively) such that X embeds densely into Y . We prove that it is consistent with ZFC that there exists a regular dense in itself countable space which is not countably connectifiable giving thus a partial answer to Problem 3.9 of Watson and Wilson (1993). On the other hand we show that Martins axiom implies that every countable dense in itself space X with πω ( X ) ω is countably connectifiable. We also establish that a separable metrizable space without open compact subsets can be densely embedded in a metric continuum.

Uniformly continuous homeomorphisms

Abstract Let (X, d) be a metric space. Under which conditions is every homeomorphism from X onto X uniformly continuous with respect to (the uniformity generated by) the metric d? We give sufficient conditions for the above question and necessary conditions for it in the case of a 0-dimensional homogeneous space. It is also proved that u.c.h.-ness for every compatible metric implies compactness for a nonrigid metrizable space. Furthermore, the interplay between u.c.h.-ness and local m-compactness is considered in the class of uniform spaces.

Maximal pseudocompact spaces and the Preiss-Simon property

We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.

On local versions of metric uc-ness, hypertopologies and function space topologies

Abstract We characterize a metric uc-ness of local nature, uc-ness means some continuity is uniform, as a uniform separation property. Then we reformulate it as a relationship between hypertopologies and finally as agreement between function space topologies.