Under the assumption that the real line cannot be covered by -many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into -many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into -many closed sets; and (c) no compact Hausdorff space can be partitioned into -many closed -sets.
Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification...
We prove that the maximal Hausdorff compactification of a -compactifiable mapping and the maximal Tychonoff compactification of a Tychonoff mapping (see [P]) are perfect. This allows us to give a characterization of all perfect Hausdorff (respectively, all perfect Tychonoff) compactifications of a -compactifiable (respectively, of a Tychonoff) mapping, which is a generalization of two results of Skljarenko [S] for the Hausdorff compactifications of Tychonoff spaces.
The following general question is considered. Suppose that is a topological group, and , are subspaces of such that . Under these general assumptions, how are the properties of and related to the properties of ? For example, it is observed that if is closed metrizable and is compact, then is a paracompact -space. Furthermore, if is closed and first countable, is a first countable compactum, and , then is also metrizable. Several other results of this kind are obtained....
We show that a Tychonoff space is the perfect pre-image of a cofinally complete metric space if and only if it is paracompact and cofinally Čech complete. Further properties of these spaces are discussed. In particular, cofinal Čech completeness is preserved both by perfect mappings and by continuous open mappings.
In this paper we introduce perfectly supportable semigroups and prove that they are σ-discrete in each Hausdorff shiftinvariant topology. The class of perfectly supportable semigroups includes each semigroup S such that FSym(X) ⊂ S ⊂ FRel(X) where FRel(X) is the semigroup of finitely supported relations on an infinite set X and FSym(X) is the group of finitely supported permutations of X.
We provide a necessary and sufficient condition for the Higson compactification to be perfect for the noncompact, locally connected, proper metric spaces. We also discuss perfectness of the Smirnov compactification.
It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes it possible...