On spaces in which countably compact sets are closed, and hereditary properties

A space X is called C-closed if every countably compact subset of X is closed in X. We study the properties of C-closed spaces. Among other results, it is shown that countably compact C-closed spaces have countable tightness and under Martin's Axiom or 2^ω0<2^ω1, C-closed is equivalent to sequential for compact Hausdorff spaces. Furthermore, every hereditarily quasi-k Hausdorff space is Fréchet-Urysohn, which generalizes a theorem of Arhangel'sk in [4]. Also every hereditarily q-space is hereditarily of pointwise countable type and contains an open dense first countable subspace.     

Direct Constructions of the Paracompact Coreflections of Frames

We present a general construction from which one can derive in a unified manner (i) the Stone–ech compactification of a frame, (ii) the Lindelöf coreflection for both frames and nearness frames and (iii) the paracompact coreflection in the category of frames.     

A normal countably compact not absolutely countably compact space

We construct an example of a normal countably compact not absolutely countably compact space. We also prove that every hereditarily normal countably compact space is absolutely countably compact and suggest a method for construction of hereditarily normal spaces without property .

     

On the Radon-Nikodym theorem for measures with values in vector lattices

Measures with values in a countably order-complete vector lattice are considered. The underlying σ-algebra is assumed to be σ-isomorphic to the Borel sets of the real line. Given one such measure, densities are searched which are not necessarily scalar-valued for smaller measures. The results can be used to prove the existence of a least upper bound for two such measures.     

Paracompact locally Lindelöf spaces and <Emphasis Type=”Italic”>L</Emphasis>-systems

Summary We establish relationships between paracompact locally Lindel&ouml;f spaces and all kinds of spaces with L-systems by means of closed mappings, weakly perfect mappings and quotient mappings. In particular, we obtain some characterizations of paracompact locally Lindel&ouml;f spaces. Interestingly, they are also characterized as weakly perfect inverse images of locally separable metric spaces.     

On Hilbert modules over locally <Emphasis Type="Italic">C</Emphasis>*-algebras II

Summary In this paper we study the unitary equivalence between Hilbert modules over a locally <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>C^{*}$-algebra. Also, we prove a stabilization theorem for countably generated modules over an arbitrary locally $C^{*}$-algebra and show that a Hilbert module over a Fr\'{e}chet locally $C^{*}$-algebra is countably generated if and only if the locally $C^{*}$-algebra of all ``compact' operators has an approximate unit.     

A metric space of A. H. Stone and an example concerning -minimal bases

In this paper we use a metric space due to A. H. Stone and one of its completions to construct a linearly ordered topological space that is \v{C}ech complete, has a -closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a -minimal base for its relative topology. However, is not metrizable and is not quasi-developable. The construction of is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone's metric space and our construction.

     

Applications of pseudo-monotone operators with some kind of upper semicontinuity in generalized quasi-variational inequalities on non-compact sets

Let be a topological vector space and be a non-empty subset of . Let and be two maps. Then the generalized quasi-variational inequality (GQVI) problem is to find a point and a point such that for all . We shall use Chowdhury and Tan's 1996 generalized version of Ky Fan's minimax inequality as a tool to obtain some general theorems on solutions of the GQVI on a paracompact set in a Hausdorff locally convex space where the set-valued operator is either strongly pseudo-monotone or pseudo-monotone and is upper semicontinuous from to the weak-topology on for each non-empty finite subset of .

     

关于完全强仿紧空间的刻画

在正则T1的条件下得到了完全强仿紧空间的一些等价刻画,并且利用获得的刻画证明了这类空间的一个可数乘积定理.     

Grasping the phantom – a new approach to nonlinear spectral theory

A new definition of a spectrum for nonlinear operators is suggested, which is called phantom. The phantom naturally divides into two subsets which in the linear case correspond to the point spectrum, and to the union of the continuous and residual spectrum. There is also a natural nonlinear analogue to the approximate point spectrum. The phantom may be considered as a variant of the spectrum of M. Furi, M. Martelli, and A. Vignoli (Ann. Mat. Pura Appl. 118 (1978), 229–294), which reflects the behavior of the operator on bounded sets and which is based on a new definition of an eigenvalue that was suggested earlier by the authors (Nonlinear Anal. 40 (2000), 565–576).?To define the phantom, the class of stably 0-epi maps is introduced and studied. This is a subclass of 0-epi maps satisfying a Rouché type theorem. Received: November 15, 1999?Published online: October 2, 2001