Some Conclusions Related To Diagonals

Compact spaces and metric spaces are important topological spaces in general topology, paracompact spaces are important generalization of them. The spaces, such as paracompact spaces, metacompact spaces and so on, which are defined by different types of open covers and their refinements are the generalization of certain property of metric spaces. Defined by such a unique way these spaces are known as covering properties, but not generalized metric spaces. However, covering properties and generalized metric spaces have a close relationship. As we know, for any set X, we say the subset ΔX={(χ,χ): χ∈X} of X x X is the diagonal of X. There are numerous results in the literature showing that the diagonal of X carries enormous information about X. Therefore, it is necessary to study spaces with certain diagonal.Let ΔX={(χ,χ): χ∈X} be the diagonal of a space X. Let A (?) X and let Y=(A×X) U (X x A). If there exists a sequence{Vn: n∈N} of open subsets of X2such that ΔX (?) Vn for all n∈N,and(∩{(?): n∈N})nY=(∩{Vn: n∈N})nY={{α, α):α∈A}, then X is said to have a regular Gδ-diagonal related to the set A. In this note, an equivalent condition of a space X which has a regular Gδ-diagonal related to a subset A of X is given. We show that if the set A is a bounded subset of a regular space X and X has a regular Gδ-diagonal related to the set A, then A is metrizable. This generalizes a conclusion which was shown by Arhangel’skii and Burke in2006. Let X be a space and F (?) X. If there is a continuous closed map f: Xâ†'[0,1] such that F=f-1(0), then the set F is a strong zero-set. Then we show that if F is a bounded strong zero-set of a regular space X and X has a regular Gδ-diagonal related to F, then F is compact and metrizable.By investigating covering properties and different separation properties of the diagonal of a space X, We give concepts of regular Δ-metacompact and functionally Δ-metacompact. A space X is regular Δ-metacompact if for every A (?) X2\ΔX closed in X2there exists a point finite open cover U of X such that A n∩(∪{U x U: U∈u})=φ. A space X is functionally Δ-metacompact if for every A (?) X2\Δx closed in X2there exists a point finite open cover U of X by functionally open sets such that A∩(∪{U x U: U∈u})=φ. Then we give a discussion on their proposities. We show that a regular Δ-matacompact space is regular. We also show that a normal Δ-matacompact space is functionally Δ-metacompact.Besides, in the third part of this paper, we mainly prove that for any closed set F C X2, ΔX (?) F, for any closed set A (?) X2with A n F=φ, if there are disjoint open sets UA and UF, such that A (?) UA, F (?) UF, then X is functionally Δ-normal. Meanwhile, it is shown that a discrete colletion of compact sets F={F_s: s∈S} of a Δ-paracompact regular space X canbe separated by a discrete open collection {W_s: s∈S} of X.