Functions In Topological Spaces Research Articles

Limit sets, continuity, and uncertainty points of mappings of topological spaces

Limit sets, continuity, and uncertainty points of mappings of topological spaces

Topological methods in the theory of operator inclusions in Banach spaces. II

We develop topological methods for the investigation of operator inclusions in Banach spaces, prove the generalized Ky Fan inequality, and study the critical points of many-valued mappings in topological spaces.

Strongly precontinuous functions

We introduce a new class of functions called strongly precontinuous functions in topological spaces. Some properties and several characterizations of this type of functions are obtained. We investigate the relationships between this class of functions and other classes of non-continuous functions.

Expansion of α‐open sets and decomposition of α‐continuous mappings

We introduce the notions of expansion 𝒜α of α‐open sets and 𝒜α‐expansion α‐continuous mappings in topological spaces. The main result of this paper is that a map f is α‐continuous if and only if it is 𝒜α‐expansion α‐continuous and ℬα‐expansion α‐continuous, where 𝒜α, ℬα are two mutually dual expansions.

On πgp-continuous functions in topological spaces

The concept of πgp-closed sets was introduced by Park [On πgp-closed sets in topological spaces, Indian J. Pure Appl. Math., in press]. The aim of this paper is to consider and characterize πgp-irresolute and πgp-continuous functions via the concept of πgp-closed sets and to relate these concepts to the classes of πGPO-compact spaces and πGP-connected spaces.

Semicontinuous utility functions in topological spaces

Rader [7] in 1963 gave a proof of a theorem about the existence of an upper semicontinuous function on a second countable topological spaceX with a total preorder\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \), provided that the left half lines\(\left\{ {x : x \prec y} \right\}\) are open for eachy belonging toX. The proof of Rader was constructive and direct, without any use of theorems like the Continuoum Characterization Theorem.

Topological intersection theorems for two set—valued mappings and applications to minimax inequalities

In this paper, we introduce sufficient conditions for the non—empty intersection of two set—valued mappings in topological spaces. As applications, some topological minimax inequalities for two functions in which one of them is separately lower (or upper) semicontinuous are given. Finally, by employing our topological intersection theorems for two set—valued mappings, some other minimax inequalities have been derived without separately lower (or upper) sernicontinuity under but with another condition. These results are topological versions of corresponding minimax inequalities for two functions due to Fan (1964) and Sion (1958) in topological vector spaces

Weakly α‐continuous functions

In this paper, we introduce the notion of weakly α‐continuous functions in topological spaces. Weak α‐continuity and subweak continuity due to Rose [1] are independent of each other and are implied by weak continuity due to Levine [2]. It is shown that weakly α‐continuous surjections preserve connected spaces and that weakly α‐continuous functions into regular spaces are continuous. Corollary 1 of [3] and Corollary 2 of [4] are improved as follows: If f1 : X → Y is a semi continuous function into a Hausdorff space Y, f2 : X → Y is either weakly α‐continuous or subweakly continuous, and f1 = f2 on a dense subset of X, then f1 = f2 on X.

Domains of Paracompactness and Local Compactness

Given a class of topological spaces and a class of mappings of topological spaces, the -résolvant of is denned to be the class of topological spaces all of whose -images lie in . Whenever is closed under composition and includes identity maps, is easily seen to be the largest class of spaces smaller than which is closed under -images.

Continuous mappings of topological spaces

Continuous mappings of topological spaces

Domains of Paracompactness and Regularity

Given a class of topological spaces and a class of maps of topological spaces, our interest is in characterization of the class () of topological spaces whose every -image lies in . The class () is referred to as the -resolvent of and is the largest class of spaces smaller than closed under -images (provided is closed under composition and includes identity maps).In the present paper, will be either the class of paracompact spaces or the class of regular spaces, and the conditions determining will always include separation axioms on the range, some of which can be dispensed with now by agreeing that all spaces are Hausdorff.

A property of irreducible images of extremally disconnected hyperstonian compacta and its application to the theory of partially ordered spaces

All topological spaces considered in this paper will be assumed to be at least Hausdorff, and mappings of topological spaces will be assumed to be continuous. It is shown that the irreducible image of an infinite hyperstonian compactum cannot be locally connected, or, in other words, that the absolute of a locally connected infinite compactum cannot be a hyperstonian compactum. As an application of this theorem it is shown, for example, that there are no nonzero completely linear functionals in the K-lineal C(S) of all continuous real functions on a locally connected compactum S without isolated points. This generalizes the known result for the case S = [0, 1], apparently first obtained by M.A. Ermolin in his candidate's thesis. In the topological part of the paper the authors basically adhere to the terminology of [1], and in the part relating to the theory of partially ordered spaces to that of [2]. Since the term component is used both in topology and in the theory of partially ordered spaces (and in different senses), we shall, to avoid confusion, speak of a component of connectivity when using it in the topological sense. Recall that a topological space is called extremally disconnected if the closure of each of its open sets is open. Recently in connection with results of Ponomarev it was shown that the class of all absolutes coincides with the class of extremally disconnected spaces (see, for example, [3]); for the case of local compacta this had been done earlier by Gleason, which increased the interest in these spaces among topologists (see, for example, [5-7]). For mathematicians working in linear partially ordered spaces, extremally disconnected spaces are very important because, as is well known (see [2], Chap. V), every archimidean K-lineal (vector lattice) X has an isomorphic representation as a K-lineal of continuous real functions on an extremally disconnected compactum (which is the Stone space of the complete Boolean algebra of components of the K-linear X), where the functions can have on nowhere dense sets the values =}=oo By a measure on a complete Boolean algebra ~ we shall understand a strictly positive countably real function p, which may take the value +oo The measure # is called locally finite if for every B ~ ~ for which # (B) = +co, there is an A~ ~, A ~ , A -< B such that 0 < ~ < +~. Note that a locally finite measure is completely additive. An extremally disconnected compactum is called hyperstonian if there exists a locally finite measure on the complete Boolean algebra of its open-closed sets.

Correction to "On real-valued functions in topological spaces"

Correction to "On real-valued functions in topological spaces"

On Real-Valued Functions in Topological Spaces

On Real-Valued Functions in Topological Spaces