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Non-trivial non weakly pseudocompact spaces
Abstract
A space Z is weakly pseudocompact if Z is Gδ-dense in at least one of its compactifications. In 1996 F.W. Eckertson [3] proposed the following problem: Find examples of Baire non Lindelöf spaces which are not weakly pseudocompact. Eckertson proposed a list of natural candidates. In this article we show that part of this list produces examples of this type by providing examples of product spaces which are Baire non-Lindelöf and not weakly pseudocompact.
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A space X is truly weakly pseudocompact if X is either weakly pseu- docompact or Lindelof locally compact. We prove: (1) every locally weakly pseu- docompact space is truly weakly pseudocompact if it is either a generalized linearly ordered space, or a proto-metrizable zero-dimensional space with (x,X) >! for every x 2 X; (2) every locally bounded space is truly weakly pseudocompact; (3) for !
In this paper we study Baire category in spaces of continuous, real-valued functions equipped with the topology of pointwise convergence. We show that, for normal spaces, the Baire category of Cπ(X) is determined by the Baire category of Cπ(X) for certain small subspaces Y of X and that the category of Cπ(X) is intimately related to the existence of winning strategies in a certain topological game Γ(X) played in the space X. We give examples of certain countable regular spaces for which Cπ(X) is a Baire space and we characterize those spaces X for which Cπ(X) has one of the stronger completeness properties, such as pseudocompleteness or Cech-completeness. © 1980, University of California, Berkeley. All Rights Reserved.
We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces C_p(X,G) of G-valued continuous functions on a space X with the topology of pointwise convergence, for a separable metric group G. Not only the results but also the proofs themselves are novel even in the classical case when G is the real line. A space X is weakly pseudocompact if it is G_delta-dense in at least one of its compactifications. A topological group G is precompact if it is topologically isomorphic to a subgroup of a compact group. We prove that every weakly pseudocompact precompact topological group is pseudocompact, thereby answering positively a question of Tkachenko.
We prove that every topological group which is a finite union of Čech-complete spaces is Čech-complete. We also show that x is Čech-complete if the product x ω is a finite union of Čech-complete subspaces. In case of finite products, we describe a model of ZFC with a non-Čech-complete space x such that x n is the union of two Čech-complete subspaces for each n∈ω.
A space X is weakly-pseudocompact if X is G δ -dense in one of its compactifications. This concept generalizes, in a very natural way, pseudocompactness. In this paper, we study weakly-pseudocompact spaces. It is shown that a space X is compact iff X is Lindelöf and weakly-pseudocompact. If X is locally compact, then X is weakly- pseudocompact iff X is either compact or it is not Lindelöf; every weakly-pseudocompact metric space is completely metrizable; and the hedgehog J(α), for ω<α, is a weakly-pseudocompact space. We introduce and study M-bounded, M-compact, for ∅≠M⊆C(X), and C B(X) -embedded subsets of a space X: these concepts are generalizations of some notions that have been considered by several people. We also prove that a space X is pseudocompact iff the set of P-points of X and Fr(Z), for each zero set of X, are bounded, and the real numbers ℝ cannot be embedded as a zero set in a pseudocompact space.
This chapter discusses cardinal functions. Cardinal functions extend important topological properties as countable base, separable, and first countable to higher cardinality. Cardinal functions allow one to formulate, generalize, and prove results of a particular type in a systematic and elegant manner. The cardinal functions also allow one to make precise quantitative comparisons between certain topological properties. Prerequisite for work in cardinal functions is knowledge of cardinal and ordinal numbers and transfinite constructions. The chapter also describes the cardinal functions on the two most important classes of abstract topological spaces, namely, compact spaces and metrizable spaces. It also describes cardinal functions that are used to obtain bounds on the cardinality of a space. An infinite cardinal that is not regular is said to be singular. Theorems from combinatorial set theory play an important role in cardinal functions.
We prove that if X is a quasiregular, countably compact space with a pseudobase consisting of closed GδGδ-sets, then every GδGδ-dense subspace of X is pseudocomplete in the sense of Todd. In particular, every weakly pseudocompact space is pseudocomplete in the sense of Todd. Some sufficient conditions are found that guarantee that a weakly pseudocompact space is pseudocomplete in the sense of Oxtoby. It is shown that every weakly pseudocompact space without isolated points has cardinality at least continuum. An example is given of a weakly pseudocompact space with one non-isolated point that contains the one-point lindelöfication of an uncountable discrete space. We apply this example to show that weak pseudocompactness is not preserved by the relation of M-equivalence.
Each open subspace of a weakly pseudocompact space is either weakly pseudocompact or locally compact Lindelöf. A topological sum is weakly pseudocompact if and only if 1.(1) each summand is either weakly pseudocompact or locally compact Lindelöf and2.(2) the sum is either compact or not Lindelöf. If X is realcompact and the lattice of compactifications of X is a b-lattice or if X is a not Čech-complete Gδ-diagonal space then X is not weakly pseudocompact. Weak pseudocompactness is neither an inverse invariant of perfect maps nor an invariant of open maps with compact fibers. There is a not pseudocompact space in which each zero set is weakly pseudocompact.