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By considering two exemplary situations, it is shown that only under rather restrictive assumptions it is, in general, possible to derive properties of the underlying topology from properties of the associated Baire and Borel σ-algebra, respectively.
Abstract
The purpose of this note is to show that there exist two Tychonoff spaces X, Y, a subset A of X and a subset B of Y such that A is weakly almost Lindelf in X and B is weakly almost Lindelf in Y, but A B is not weakly almost Lindelf in X Y.
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We show that a monotonically normal space X is paracompact if and only if for every increasing open cover {U α : α < κ} of X, there is a closed cover {F nα : n < ω, α < κ} of X such that F nα ⊂ U α for n < ω, α < κ and F nα ⊂ F nβ if α ≦ β.
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Some characterizations of S-paracompact spaces are given. We introduce a class of S-expandable spaces and study topological properties of S-expandable spaces.
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We show that every T 1 wN-space is expandable and as a corollary, we prove that a ϑ-refinable sym-wg T 1 space is paracompact and thus two problems of Chris Good are solved. We also investigate s-expandability, and for extremally disconnected spaces, a characterization of s-expandability is given in terms of covers, which gives an extension to a known result.
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We prove that if a space X is well ordered (αA), or linearly semi-stratifiable, or elastic then X is a D-space.
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A number of generalized metric spaces have been defined or characterized in terms of g-functions. Symmetric g-functions are discussed by C. Good, D. Jennings and A. M. Mohamad. In this paper, some questions about symmetric g-functions are answered, particularly it is shown that every sym-wg-space is expandable.
Abstract
By introducing a conservative notion of functional compactness to frames and using frame-theoretic techniques, we identify two classes of topological spaces that can be densely embedded in functionally compact ones. Since θ-open elements of frames (corresponding in spaces to complements of Veličko’s [15] θ-closed subspaces) play an important role in functional compactness, we also study the concept of co-rigidity in frames and show that it is enjoyed by all θ-open elements of functionally compact frames.
Abstract
A space (X, T) is called I-Lindelöf [1] if every cover A of X by regular closed subsets of the space (X, T) contains a countable subfamily A′ such that X = ∪{int (A): A ∈ A′}. In this work we introduce the class of I-Lindelöf sets as a proper subclass of rc-Lindelöf sets [3]. We study various properties of I-Lindelöf sets and investigate the relationship between I-Lindelöf sets and I-Lindelöf subspaces. We give a new characterization of I-Lindelöf spaces in terms of this type of sets. Also, we study spaces (X, T) in which every I-Lindelöf set in (X, T) is closed.
