When a compact (countably compact) set is closed. II

We obtain (a) necessary and sufficient conditions and (b) sufficient conditions for a compact (countably compact) set to be closed in products (sequential products) and subspaces (sequential subspaces) of normal spaces. As a consequence of these, sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets in these spaces. It is also shown that these results do not hold for quotients of even T ₄,-spaces.     

Continuous selections as parametrically defined integrals

An analog of the classical Michael theorem on continuous single-valued selections of lower semicontinuous maps whose values are closed and convex in a Fréchet space is proved for maps into metrizable (non-locally-convex) vector spaces. It turns out that, instead of the local convexity of the whole space containing these values, it is sufficient to require that the family of values of the map be uniformly locally convex. In contrast to the standard selection theorems, the proof bypasses the process of successively improving the approximations, and the desired selection is constructed as the result of pointwise integration with respect to a suitable probability distribution.     

亚紧性在选择理论中的一个刻划(英文)

本文利用集值映射的选择理论,对正则空间的亚紧性给予了一种刻划。本文同时推广了选择理论中的两个经典定理。     

相对拓扑的一些性质

讨论了相对超正则的充要条件及相对拓扑空间在映射下的一些性质，并定义了两类新的相对拓扑空间，给出了它们的一些性质．     

Monotone normality and paracompactness

In this paper it is proved that a topological space is necessarily paracompact if it is monotonically normal and any one of the following: screenable, paralindelöf, has a G_δ-diagonal or a quasi-G_δ-diagonal or has a σ-locally-countable base.     

Remarks on discretely absolutely star-Lindelöf spaces

In this paper, we prove the following statements(1) There exists a Hausdorff Lindelöf space X such that the Alexandroff duplicate A(X) of X is not discretely absolutely star-Lindelöf.(2) If X is a regular Lindelöf space, then A(X) is discretely absolutely star-Lindelöf.(3) If X is a normal discretely star-Lindelöf space with e(X) < ω ₁, then A(X) is discretely absolutely star-Lindelöf.     

On expansions and extensions of powerful digraphs

We study the problem of expanding and extending the structure of a stable powerful digraph to the structure of a stable Ehrenfeucht theory. We define the concepts of type unstability and type strict order property. We establish the presence of the type strict order property for every acyclic graph structure with an infinite chain. The simplest form of expansion of a powerful digraph to the structure of an Ehrenfeucht theory is the expansion with a 1-inessential ordered coloring and locally graph ?-definable many-placed relations, which enable us to mutually realize nonprincipal types; we prove that this expansion is incapable of keeping the structure in the class of stable structures, and moreover, by the type strict order property it generates the first-order definable strict order property. We define the concept of a locally countably categorical theory (LCC theory) and prove that given the list p ₁(x), ..., p _n (x) of all nonprincipal 1-types in an LCC theory, if all types r(x ₁, ..., x _m ) containing \(p_{i_1 } \) (x ₁) ∪ ... ∪ \(p_{i_m } \)(x _m ) are dominated by some type q then q is a powerful type.     

Generalized Quasi—Variational Inequalities on Paracompact Sets

Ｔｈｉｓｐａｐｅｒｉｓｃｏｎｔｉｎｕａｔｉｏｎｓｏｆ［１］，ｗｅｓｔｉｌｌｔｏｕｓｅｍａｒｋｓａｎｄｔｅｒｍｓｉｎ［１］ａｎｄｔｈｅｏｔｈｅｒｔｅｒｍｓａｇｒｅｅｗｉｔｈ［２］ａｎｄ［３］．ＬｅｔＥａｎｄＦｂｅｔｗｏＨａｕｓｄｏｒｆｆｔｏｐｏｌｏｇｉｃａｌｖｅｃｔｏｒｓｐａｃｅｓ，ＸＥ，ＹＦｂｅｔｗｏｎｏｎｅｍｐｔｙｓｅｔｓ，ＦｂｅｔｈｅｄｕａｌｓｐａｃｅｏｆＦ，Ａ：Ｘ→２ＹａｎｄＢ：Ｙ→２Ｆｂｅｔｗｏｓｅｔ－ｖａｌｕｅｄｍａｐｐｉｎｇ，Ｔ：Ｙ→Ｘｂｅｉｎｖｅｒｔｉｂｌｅ．Ｉｎｔｈｉｓｐ…     

On the use of purely finitely additive multipliers in mathematical programming

In this paper, duality results are obtained for the problem of finding =inf{f(x):g(x)0,g(x)Y,xC}, where the constrainst spaceY is someL -space, by using the norm topology ofY. The corresponding multiplier is the sum of a countably additive part and a purely finitely additive part. Conditions are given such that the purely finitely additive part may be discarded.     

实值函数与某些覆盖性质

In the paper [properties defined with semi-continuous functions and some related spaces',Houston J.Math.,2015,41(3):1097–1106] properties(U L)~(wl)_m,(U L)~K_m and(U L)_m were defined and it was shown that spaces having these properties coincide with countably paracompact spaces,countably mesocompact spaces and countably metacompact spaces,respectively.In this paper,we continue with the study on the relationship between properties defined with real-valued functions and some covering properties.Some characterizations of countably compact spaces and pseudo-compact spaces in terms of real-valued functions are obtained.