Continuous Real-valued Functions Research Articles

On Some Topological Properties of Ch(X) and of the Dual Operator

The hypo-topology on the algebra C(X) of real-valued continuous functions defined on a Tychonoff space 2X f ∈ C(X) with its hypograph, hypof = {(x,t) ∈ X x R:f(x) ≥ t}. This topology is very useful in the calculus of variations and in optimization theory (e.g. Maximization problems). We denote C(X) with the hypo-topology by Ch(X). Our study deals with fundamental properties of these function spaces, and with the linear operators on them and as well as the characterization of the topological properties of Ch(X) in terms of topological properties of the base space X. We are studying the linear operator between the functional algebras Ch(X) and the Ch(Y). We are primarily concerned with the continuity of the evaluation functional, the general evaluation and the continuity of the characters of Ch(X) before the investigation of the properties of the dual operator f* of a continuous function f:X→Y. This operator is defined by f*:Ch (Y)→Ch (X), where f*(g)=gof for all g ∈ C(Y). The continuity of f* enables us to characterize the continuity of an algebra homomorphism of the type φ:Ch (Y)→Ch (X) for a realcompact space Y. For such a space, we present a type of Riesz theorm with states that an algebra homomorphism φ: Ch(Y)→Ch(X) is continuous if and only if there exists a unique hypo-function f: X→Y such that φ = f*. There after, we give the equivalence between the properties of f and those of f*. The study of the continuous linear functional on Ch(X) helps us to compute the topological dual space of this algebra. We show here that this dual space is useful only when the set of isolated points of X is dense.

On Ψω-factorizable groups

A topological group G is called Ψω-factorizable (resp. M-factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a topological group H with ψ(H)≤ω (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of Ψω-factorizable groups. It is shown that a topological group G is Ψω-factorizable if and only if every continuous real-valued function on G is Gδ-uniformly continuous, if and only if for every cozero-set U of G, there exists a Gδ-subgroup N of G such that UN=U. Sufficient conditions on the Ψω-factorizable group G to be M-factorizable are that G is τ-fine and τ-steady for a cardinal τ.

Comparing functional countability and exponential separability

A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponential separability. We start by studying these properties in GO spaces, where we extend results by Tkachuk and Wilson, and prove a conjecture by Dow. We also study some subspaces of products that are functionally countable and the influence of the Gδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_\\delta $$\\end{document}-topology on exponential separability. Finally, we give some examples of functionally countable spaces that are separable and uncountable.

Velichko's notions close to sequential separability and their hereditary variants in Cp-theory

A space X is sequentially separable if there is a countable S⊂X such that every point of X is the limit of a sequence of points from S. In 2004, N.V. Velichko defined and investigated concepts close to sequential separability: σ-separability and F-separability. The aim of this paper is to study σ-separability and F-separability (and their hereditary variants) of the space Cp(X) of all real-valued continuous functions, defined on a Tychonoff space X, endowed with the pointwise convergence topology. In particular, we proved that σ-separability coincides with sequential separability. Hereditary variants (hereditarily σ-separability and hereditarily F-separability) coincide with Fréchet–Urysohn property in the class of cosmic spaces.

Phase-isometries between the positive cones of the Banach space of continuous real-valued functions

Phase-isometries between the positive cones of the Banach space of continuous real-valued functions

Optimal transport for types and convex analysis for definable predicates in tracial W⁎-algebras

We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on Cn is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function Cn→Cn is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by “C1 definable predicates” whose gradients are definable functions. As a consequence, we show that every element in the definable closure of W⁎(x1,…,xn) can be expressed as a definable function of (x1,…,xn). We give several classes of examples showing that the definable closure can be much larger than W⁎(x1,…,xn) in general.

Computably and punctually universal spaces

We prove that the standard computable presentation of the space C[0,1] of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s.We prove that the original Urysohn's construction of the universal separable Polish space U is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space 2ω; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.

On the product of almost discrete Grothendieck spaces

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product X=∏Xi of almost discrete spaces Xi the space Cp(X) of all continuous real-valued functions with the topology of pointwise convergence is a μ-space if, and only if, X is a weak q-space if, and only if, t(X)=ω if, and only if, X is functionally generated by the family of all its countable subspaces.This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC, obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that X×Y is not weakly Grothendieck space. In (PFA): the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.

DISCONTINUOUS HOMOMORPHISMS OF WITH

AbstractAssume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $ , into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$ , the algebra of continuous real-valued functions on X.

On absolutely C-embedded topological groups

A topological group H is called absolutely C-embedded if H is C-embedded into every topological group containing H as a subgroup. It is shown that a left uniformly continuous real-valued function on a subgroup of a topological group G can extend to a continuous function on G. Therefore, fine groups are absolutely C-embedded. We also prove that a CM-factorizable groups are absolutely C-embedded. Additionally, we explore the factorizability of absolutely C-embedded groups and establish that an absolutely C-embedded group G is R-factorizable if and only if G is an ω-narrow group. We also show that a non-metrizable absolutely C-embedded group G is M-factorizable if and only if G is τ-precompact, where τ=Fi(G).

MONETARY UTILITY FUNCTIONS ON Cb(X) SPACES

We will characterize robust monetary utility functions defined on the space of real valued (bounded) continuous functions on a Polish space.

The Banach–Mazur distance between isomorphic spaces of continuous functions is not always an integer

For over half a century, Aleksander Pełczyński's question, whether the Banach–Mazur distance between any two isomorphic C(K) spaces is an integer, has remained open; as usual, K is a compact Hausdorff space and C(K) denotes the Banach space of all continuous real-valued functions on K, provided with the maximum norm. We answer this question in the negative. Moreover, we prove that the Banach–Mazur distance between isomorphic spaces C(K) and C0(L) is not always an integer; here L is a locally compact but noncompact Hausdorff space and C0(L) denotes the Banach space of all continuous real-valued functions vanishing at infinity on L, furnished with the maximum norm.

A multiplicity result in finite-dimensional vector spaces

Let f 1 , … , f n be n ( n ≥ 2 ) continuous real-valued functions on R such that lim | t | → + ∞ ∫ 0 t f k ( s ) d s t 2 = − ∞ for all k = 1 , … , n . This sole condition is far from ensuring the existence of multiple solutions for the classical problem { − ( x k + 1 − 2 x k + x k − 1 ) = f k ( x k ) k = 1 , … , n , x 0 = x n + 1 = 0. However, as a by-product of a much more general result, we get the following: for each ρ ∈ R and for each i = 1 , … , n , there exists ( λ , μ ) ∈ R 2 such that the problem { − ρ ( x k + 1 − 2 x k + x k − 1 ) = f k ( x k ) k = 1 , … , n , k ≠ i − ρ ( x i + 1 − 2 x i + x i − 1 ) = f i ( x i ) + λ x i + μ x 0 = x n + 1 = 0 has at least three solutions.

Compact Resolutions and Analyticity

We consider the large class G of locally convex spaces that includes, among others, the classes of (DF)-spaces and (LF)-spaces. For a space E in class G we have characterized that a subspace Y of (E,σ(E,E′)), endowed with the induced topology, is analytic if and only if Y has a σ(E,E′)-compact resolution and is contained in a σ(E,E′)-separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class G. The mentioned characterization follows from the following analogous result: The space C(X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology ξ stronger or equal than the pointwise topology τp of C(X) is analytic iff (C(X),ξ) is separable and is covered by a compact resolution.

On bounded sets in [formula omitted

In this note we study some topological properties of the bounded sets of the locally convex space Ck(X) of all real-valued continuous functions defined on a Tychonoff space X equipped with the compact-open topology as well as of the bounded sets of a (DF)-space. We show that, assuming X is a Lindelöf Čech-complete space, if X is scattered then each bounded set in Ck(X) is a Fréchet-Urysohn space under the weak topology. In the opposite direction, if X is pseudocompact and each bounded set in Ck(X) is Fréchet-Urysohn under the weak topology, then X is k-scattered. We also derive a new condition, related to the closed bounded sets, for a (DF)-space to be quasibarrelled, which we apply to the bounded sets of Ck(X). Finally, we get a sort of Krein-like theorem for Ck(X) when X is a Lindelöf Σ-space.

Stability analysis through the Bielecki metric to nonlinear fractional integral equations of $ n $-product operators

<abstract><p>This work is devoted to the analysis of Hyers, Ulam, and Rassias types of stabilities for nonlinear fractional integral equations with $ n $-product operators. In some special cases, our considered integral equation is related to an integral equation which arises in the study of the spread of an infectious disease that does not induce permanent immunity. $ n $-product operators are described here in the sense of Riemann-Liouville fractional integrals of order $ \sigma_i \in (0, 1] $ for $ i\in \{1, 2, \dots, n\} $. Sufficient conditions are provided to ensure Hyers-Ulam, $ \lambda $-semi-Hyers-Ulam, and Hyers-Ulam-Rassias stabilities in the space of continuous real-valued functions defined on the interval $ [0, a] $, where $ 0 < a < \infty $. Those conditions are established by applying the concept of fixed-point arguments within the framework of the Bielecki metric and its generalizations. Two examples are discussed to illustrate the established results.</p></abstract>

Unitary groups, $\boldsymbol{K}$ -theory, and traces

AbstractWe show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$ -theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).

Linear homeomorphisms of function spaces and the position of a space in its compactification

Abstract An old question of Arhangel’skii asks if the Menger property of a Tychonoff space X is preserved by homeomorphisms of the space $C_p(X)$ of continuous real-valued functions on X endowed with the pointwise topology. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of function spaces $C_p(X)$ by looking at the way X is positioned in its (Čech–Stone) compactification.

When C(X) is an h-local ring

An h-local domain is a domain for which each nonzero prime ideal is contained in a unique maximal ideal and each nonzero element has finite character. In his dissertation, A. Omairi generalizes the notion of an h-local domain to rings with zero-divisors by restricting the definition to regular ideals. In this article, we give a characterization of when C(X) the ring of continuous real-valued functions on a space X is an h-local ring. We are left with more questions than answers.

Ideal approach to convergence in functional spaces

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy [Topology Appl. 14 (1982), pp. 151–161], which was later reposed by A. Miller, T. Orenshtein, and B. Tsaban. Namely, we show that under p = c \mathfrak {p}=\mathfrak {c} there is a δ \delta -set that is not a γ \gamma -set. Thus we constructed a subset A A of reals such that the space C p ( A ) \mathrm {C}_p(A) of all real-valued continuous functions on A A is not Fréchet–Urysohn, but possesses the Pytkeev property. Moreover, under C H \mathbf {CH} we construct a π \pi -set that is not a δ \delta -set solving a problem by M. Sakai. In fact, we construct various examples of δ \delta -sets that are not γ \gamma -sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Fréchet–Urysohn property for many different Borel ideals in the realm of functional spaces.