Neighborhood Assignments And Rank Of Diagonals

In the first chapter of this paper, we study D-spaces and some gen-eralized D-spaces. The main results obtained are the followings:Theorem 0.0.1:IfΧ=∪i=1kΧi, where k is a natural number andΧi is a strongly∑-space, thenΧis a D-space.Theorem 0.0.2:IfΧis a finite union ofδθ-refinable spacesΧi,i= 1,...,k,thenΧis an aD-space.Theorem 0.0.3:IfΧis discrete complete and the countable union of dually-discrete spaces, thenΧis a compact space.Theorem 0.0.1 gives a positive answer to the following question posed by Arhangel'skii:Is a finite union of Moore spaces a D-space?In the second chapter, by studying the properties of symmetric neighborhood assignments, we obtain the following two results which solve two open problems posed by J.Nagata on symmetric neighborhood assignments:Theorem 0.0.4:Let X be a metrizable space. Then for each neighborhood assignment{U(x):x∈Χ} onΧ, which is uniform for some compatible metric ofΧ, there is a point-finite or point-countable symmetric neighborhood assignment{V(x):x∈Χ}, such that V(x)(?) U(x) for every x∈ΧiffΧis a strongly paracompact space.Theorem 0.0.5:A regular spaceΧis an orthocompact Moore space iffΧhas a sequence of interior-preserving symmetric neighborhood assignments Un,n∈N, such that{Un(x):n∈N} is a base for each x∈Χ.In the third chapter, we focus on the rank of diagonals and obtain the following results: Theorem 0.0.6:For each k∈ωwith k≥4, there exists a sep-arable, nonsubmetrizable and subparacompact Tychonoff space X such that the rank of the diagonal of X is k.Theorem 0.0.7:If X is a regular star-compact space with a k-in-countable base for some k∈ω, then X is metrizable.Theorem 0.0.6 not only proves a conjecture rasied by A.V.Arhangel'-skii and R.Z.Buzyakova on the rank of diagonals but also gives answers to two related open problems raised by them.