On complete partially ordered sets and compatible topologies

We describe two complete partially ordered sets which are the intersection of complete linear orderings but which have no compatible Hausdorff topology. One is two-dimensional, while the second is countable, and leads to an example of a countable, compact, T ₁ space with a countable base which is not the continuous image of any compact Hausdorff space.     

A lattice-theoretic approach to arbitrary real functions on frames

In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on L. This approach mimicks the situation one has with a T₁-space X, where the lattice F?(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T₁-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T₀-spaces.     

Ideals associated with realcompactness in pointfree function rings

Abstract

Let RL denote the ring of continuous real-valued functions on a com- pletely regular frame L. The support of an α ∈ RL is the closed quotient ↑(coz α)?. We show that if supports are coz-quotients in L, then the set of functions with realcompact support is an ideal. If L satisfies the stronger condition that supports are C-quotients, then this ideal is the intersection of pure parts of the free maximal ideals of RL. The set of functions whose cozeroes are realcompact is always an ideal, which is free if and only if L is locally realcompact if and only if L is (isomorphic to) an open quotient of υL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only if υL → L is a one-point extension of L.     

On the parallel between normality and extremal disconnectedness

Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore be studied in parallel. This paper investigates the source of this ‘duality’ and shows that each pair of parallel results can be framed by the ‘same’ proof. The key tools for this purpose are relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed class of complemented sublocales of the given locale) that bring and extend to locale theory a variety of well-known classical variants of normality and upper and lower semicontinuities in an illuminating unified manner. This approach allows us to unify under a single localic proof all classical insertion, as well as their corresponding extension results.     

General insertion and extension theorems for localic real functions

In this paper we investigate localic real functions on frames. We provide a necessary and sufficient condition for the insertion of a continuous localic real function between two arbitrary comparable localic real functions. We also establish necessary and sufficient conditions for extending a bounded localic real function from a complemented sublocale to the whole frame.     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

Concerning some variants of C-embedding in pointfree topology

We study three types of quotient maps of frames which are closely related to C- and C^?-quotient maps. We call them C₁-, strong C₁-, and uplifting quotient maps. C₁-quotient maps are precisely those whose induced ring homomorphisms contract maximal ideals to maximal ideals. We show that every homomorphism onto a frame is a C₁-, a strong C₁-, or an uplifting quotient map iff the frame is pseudocompact, compact, or almost compact and normal, respectively. These quotient maps are used to characterize normality and also certain weaker forms of normality in a manner akin to the characterization of normal frames as those for which every closed quotient map is a C-quotient map. Under certain conditions, we show that the Stone extension of a quotient map is C₁-, strongly C₁- or uplifting if the map has the corresponding property.     

Monotone insertion and monotone extension of frame homomorphisms

The purpose of this paper is to introduce monotonization in the setting of pointfree topology. More specifically, monotonically normal locales are characterized in terms of monotone insertion and monotone extensions theorems.     

On the tychonoff functor and w-compactness

The main purpose of this paper is to settle the following problem concerning a product formula for the Tychonoff functor τ, by introducing the notion of w-compact spaces: Characterize a topological space X such that τ(X×Y)=τ(X)×τ(Y) for any topological space Y. We also study the properties of w-compact spaces, and it is proved that, for any family {X_α} of w-compact spaces, the product ΠX_α is also w-compact and τ(ΠX_α)=Πτ(X_α).     

The evaluation identification in function spaces

A study is made of the natural function which maps each point x of a space X to the evaluation function e_x:Y^x→Y defined by $\text{e}{}_{\mathrm{x}}\text{(?)=?(x)}$. A consequence of the results is that βX and υX can both be considered as subspaces of spaces of continous functions from appropriate domain spaces into I or R, respectively.     

On Directed Sets and their Suprema

The aim of this paper is to investigate partially ordered real linear topological spaces in which directed sets admit a supremum in their closure. In particular, we point out that this property is intimately related to the normality of the ordering cone and also to the Scott continuity of functionals belonging to the nonnegative polar of the ordering cone. Research of Mohamed Ait Mansour was supported by LACO (Laboratoire d'Arithmétique, Calcul Formel et Optimisation), UMR-CNRS 6090, University of Limoges and Agence Universitaire de la Francophonie.     

A new look at localic interpolation theorems

This paper presents a new treatment of the localic Katětov-Tong interpolation theorem, based on an analysis of special properties of normal frames, which shows that it does not hold in full generality. Besides giving us the conditions under which the localic Katětov-Tong interpolation theorem holds, this approach leads to a especially transparent and succinct proof of it. It is also shown that this pointfree extension of Katětov-Tong theorem still covers the localic versions of Urysohn's Lemma and Tietze's Extension Theorem.     

The order of starlikeness of some classes of strongly starlike functions

In this paper we consider the order of strongly starlikeness for functions in Silverman’s class 𝒢_b and in the generalized Mocanu’s class M (α, β1, β2). Some previous results are improved.     

Sigma-compact,nowhere locally compact metric spaces can be densely imbedded in Hilbert space

It is shown that if H is a connected, locally contractible, separable, topologically complete metric space with the property that mappings of separable metric spaces into H are approximable by imbeddings (in particular, if H is Hilbert space), then every sigma-compact, nowhere locally compact metric space can be densely imbedded in H.     

On Compactness of Frames

We define the notions of extension-closed and nearly closed sublocales of frames and present, amongst other things, a pointfree and choice-free proof of a result of Harris characterizing extension-closed subspaces. We also characterize compact frames in terms of the latter form of closedness.     

On sieve-complete and compact-like spaces

It is shown that a completely regular space X is sieve-complete (or, equivalenty, X is the open image of a paracompact ?ech-complete space) iff βX?X is compact-like, i.e., Player I has a winning strategy in the topological game G(C, βX?X) of [13].     

Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,II

A topological spaceX whose topology is the order topology of some linear ordering onX, is called aninterval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called aCO space and a space isscattered if every non-empty subspace has an isolated point. We regard linear orderings as topological spaces, by equipping them with their order topology. IfL andK are linear orderings, thenL ^*, L+K, L · K denote respectively the reverse ordering ofL, the ordered sum ofL andK and the lexicographic order onL x K (so · 2=+). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , l 0, letL(K,)=K+1+*.Theorem: Let X be a compact interval scattered space. Then X is a CO space if and only if X is homeomorphic to a space of the form +1+_{1 L(K i i), where is any ordinal, n , for every ii},_i are regular cardinals and K_ii, and if n>0, then max({K_i:i相似文献