A Note on the Comparison of Topologies

A considerable problemof some bitopological covering properties is the bitopological unstability with respect to the presence of the pairwise Hausdorff separation axiom. For instance, if the space is RR-pairwise paracompact, its two topologies will collapse and revert to the unitopological case. We introduce a new bitopological separation axiom τS2σ which is appropriate for the study of the bitopological collapse. We also show that the property that may cause the collapse is much weaker than some modifications of pairwise paracompactness and we generalize several results of T. G. Raghavan and I. L. Reilly (1977) regarding the comparison of topologies. 2000 Mathematics Subject Classification. 54E55, 54A10. 1. Preliminaries. The term space is referred to as a set with one, two, or three topologies, depending on the context. Let X be a set with three topologies τ , σ , and ρ. Recall that X is said to be (τ−σ) paracompact with respect to ρ (see [6]) if every τ-open cover of X has a σ -open refinement which is locally finite with respect to the topology ρ. We say that x ∈ X is a (σ ,ρ)-θ-cluster point (see [4]) of a filter base Φ in X if for every V ∈ σ such that x ∈ V and every F ∈ Φ, the intersection F∩clρ V is nonempty. If Φ has a cluster point with respect to the topology τ , we say that Φ has a τ-cluster point. Recall that X is called (τ,σ ,ρ)-θ-regular if X satisfies any of the following equivalent conditions (see [4]): (i) For every τ-open cover Ω of X and each x ∈X there is a σ -open neighborhood U of x such that clρ U can be covered by a finite subfamily of Ω. (ii) Every τ-closed filter base Φ with a (σ ,ρ)-θ-cluster point has a τ-cluster point. (iii) Every filter base Φ with a (σ ,ρ)-θ-cluster point has a τ-cluster point. (iv) For every filter base Φ in X with no τ-cluster point and every x ∈ X there are U ∈ σ , V ∈ ρ, and F ∈ Φ such that x ∈U , F ⊆ V , and U∩V =∅. For example, a space which is (τ−ρ) paracompact with respect to σ is (τ,σ ,ρ)θ-regular. Let (X,τ,σ) be a bitopological space. Recall that X is said to be τR0 if the topology τ is R0, that is, if x ∈ U ∈ τ implies clτ{x} ⊆ U . We say that the space X is τRσ (see [6]) if for each x ∈ X, clτ{x} = {V | V ∈ σ, x ∈ V} ={clτ V | V ∈ σ, x ∈ V}. Recall that the space X is called τ locally compact with respect to σ (see [5]) if each point x ∈ X has a neighborhood in σ , which is compact in τ . We say that X is τ paracompact with respect to σ (see [5]) if each τ-open cover of X has a τ-open refinement which is locally finite with respect to σ , that is, if X is (τ−τ) paracompact with respect to σ . The spaceX is said to be τ α-paracompact with respect to σ (see [6]) if every τ-open cover ofX admits a (τ∨σ)-open refinement which is locally finite with respect to σ . The bitopological space X is said to be α-pairwise paracompact (see [6])