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In a topological spaceX, a T2-distinct pointx means that for anyy∈X x≠y, there exist disjoint open neighbourhoods ofx andy. Similarly, T0-distinct points and T1distinct points are defined. In a Ti-distinct point-setA, we assume that eachx∈A is a T i -distinct point (i=0, 1, 2). In the present paper some implications of these notions which ‘localize’ the T i -separation axioms (i=0, 1, 2) requirement, are studied. Suitable variants of regularity and normality in terms of T2-distinct points are shown hold in a paracompact space (without the assumption of any separation axioms). Later T0-distinct points are used to give two characterizations of the R D -axiom.1 In the end, some simple results are presented including a condition under which an almost compact set is closed and a result regarding two continuous functions from a topological space into a Hausdorff space is sharpened. A result which relates a limit pointv to an ω-limit point is stated.
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We show that there are exactly five different classes of complete regularity determined by finite topological spaces.
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This note is devoted to study the preservation of connectedness under the basic operators in generalized topological spaces. Some characterizations of generalized connectedness are given. As an application, we generalize some results in topological spaces.
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Extremal disconnectedness is further investigated for generalized topological spaces. It is found that extremally disconnected generalized topological spaces are a rich source of generalized lower semi-continuous and generalized upper semi-continuous mappings.
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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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We study the relationship between the product and other basic operations (namely σ, π, α and β) of generalized topologies. Also we discuss the connectedness, generalized connectedness and compactness of products of generalized topologies. It is proved that the connectedness and compactness are preserved under the product of generalized topologies, which shows that the definition of product of generalized topologies is quite reasonable.
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We define and study complete generalized neighborhood systems, and prove that a generalized neighborhood system is complete if and only if it can be generated by a generalized topology. Also we obtain some applications of complete generalized neighborhood systems.
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A classical insertion theorem due to Katětov–Tong (or Dowker–Katětov, Michael) reads that ℝ can be a test space for the range of maps on the insertion theorem which characterizes the domain to be normal (or normal and countably paracompact, perfectly normal). It is known that the range ℝ in the Katětov–Tong insertion theorem is not necessarily replaced by a non-trivial separable Banach lattice. We show that the range ℝ in the Dowker–Katětov and Michael insertion theorems can be replaced by any non-trivial separable Banach lattice.
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We give a systematic discussion on the relationship among generalized topologies, generalized neighborhood systems, and generalized interior operators. As some applications, we answer a question raised in [7] by Shen, and characterize generalized continuous maps.
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The concept of normality is defined for generalized topologies in the sense of [1], a few properties of normal spaces are proved, and their characterization with the help of a suitable form of Urysohn’s lemma is discussed.
