Continuous Real-valued Functions Research Articles

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.

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, -theory, and traces

Abstract We 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.

A Chebyshev-type alternation theorem for best approximation by a sum of two algebras

AbstractLet X be a compact metric space, C(X) be the space of continuous real-valued functions on X and $A_{1},A_{2}$ be two closed subalgebras of C(X) containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_{1}+A_{2}$. We prove a Chebyshev-type alternation theorem for a function $u_{0} \in A_{1}+A_{2}$ to be a best approximation to f.

On subspaces of spaces C_p(X) isomorphic to spaces c_{0} and ell _{q} with the topology induced from mathbb {R}^{mathbb {N}}

The linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space mathbb {R}^X) is denoted by C_p(X). In this paper we continue the systematic study of sequences spaces c_{0} and ell _{q} (for 0<qle infty ) with the topology induced from mathbb {R}^{mathbb {N}} (denoted by (c_{0})_p and (ell _{q})_{p}, respectively) and their role in the theory of C_p(X) spaces. For every infinite Tychonoff space X we construct a subspace F of C_p(X) that is isomorphic to (c_{0})_p; if X contains an infinite compact subset, then the copy F of (c_{0})_p is closed in C_p(X). It follows that C_p(X) contains a copy of (ell _{q})_{p} for every 0<qle infty . We prove that for any infinite compact space X the space C_p(X) contains no closed copy of (ell _{q})_{p} for qin (0, 1]cup {infty } and no complemented copy for 0<qle infty . Relation with results of Talagrand, Haydon, Levy and Odell will be also discussed. Examples and open problems will be provided.

Zero-divisor graph and comaximal graph of rings of continuous functions with countable range

In this paper, two outwardly different graphs, namely, the zero-divisor graph [Formula: see text] and the comaximal graph [Formula: see text] of the ring [Formula: see text] of all real-valued continuous functions having countable range, defined on any zero-dimensional space [Formula: see text], are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness are concerned. However, the study reveals that the zero-divisor graph [Formula: see text] of an intermediate ring [Formula: see text] of [Formula: see text] is complemented if and only if the space of all minimal prime ideals of [Formula: see text] is compact. Moreover, [Formula: see text] is complemented when and only when its subgraph [Formula: see text] is complemented. On the other hand, the comaximal graph of [Formula: see text] is complemented if and only if the comaximal graph of its over-ring [Formula: see text] is complemented and the latter graph is known to be complemented if and only if [Formula: see text] is a [Formula: see text]-space. Indeed, for a large class of spaces (i.e. for perfectly normal, strongly zero-dimensional spaces which are not P-spaces), [Formula: see text] and [Formula: see text] are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilized to establish that for a discrete space [Formula: see text], [Formula: see text] (= [Formula: see text]) and [Formula: see text] (= [Formula: see text]) are isomorphic, if [Formula: see text] is at most countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.

On bM-ω-balancedness and [formula omitted]-factorizability of para(semi)topological groups

In this article, we introduce a notion which is called bM-ω-balancedness for semitopological groups. We show that a semitopological group G is topologically isomorphic to a subgroup of the product of a family of metrizable semitopological groups if and only if G is T0bM-ω-balanced. Thus a semitopological group G is T0bM-ω-balanced if and only if G is regular, has property (⁎) and countable index of regularity. The notion of property (⁎) is in the sense of I. Sánchez in [16].We show that if G is a T0bM-ω-balanced weakly Lindelöf paratopological group with a q-point, then for every continuous real-valued function f:G→R, there exist a continuous clopen homomorphism π:G→H onto a separable metrizable paratopological group H such that ker(π) is countably compact and a continuous real-valued function g:H→R such that f=g∘π.We introduce the notions of M (DE)-factorizability for semitopological groups. We show that a semitopological group G is Tychonoff and M-factorizable if and only if G is T0bM-ω-balanced and has property ω-QU. If G is a Tychonoff M-factorizable semitopological group and f:G→H is an open continuous homomorphism onto a semitopological group H such that ker(f) is countably compact, then H is DE-factorizable.

Nonlinear disjointness/supplement preservers of nonnegative continuous functions

Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying thatf∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such thatΦ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y.Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C0(X)+→C0(Y)+ (resp. Φ:Cb(X)+→Cb(Y)+) is a surjective map. We show that Φ preserves the norms of infima, i.e.,‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C0(X)+ (resp.Cb(X)+), if and only if there is a homeomorphism τ:Y→X such thatΦ(f)(y)=f(τ(y)),∀f∈C0(X)+ (resp.Cb(X)+), ∀y∈Y.

Real-boards of ideals in C(X) and their applications to characterize some spaces

In this paper, using the concept of the real-boards of ideals in C(X), the ring of all real-valued continuous functions on a completely regular Hausdorff space X, some characterizations of nearly pseudocompact and nearly realcompact spaces are given. For an ideal I⊆C(X), the real-board of I is the largest subspace of βX, the Stone-Čech compactification of X, on which every member of I can be extended continuously. Using this concept, the subrings CKI(X) and C∞I(X) of C(X) are introduced and studied, and I-pseudocompact spaces which are generalizations of pseudocompact spaces are presented as well as some of their properties.

Graphs of continuous functions and fractal dimensions

In this paper, we show that, for any β∈[1,2], a given strictly positive (or strictly negative) real-valued continuous function on [0,1] whose graph has the upper box dimension less than or equal to β can be decomposed as a product of two real-valued continuous functions on [0,1] whose graphs have upper box dimensions equal to β. We also obtain a formula for the upper box dimension of every element of a ring of polynomials in a finite number of continuous functions on [0,1] over the field R.

Feral dual spaces and (strongly) distinguished spaces C(X)

Following Dieudonné and Schwartz a locally convex space is distinguished if its strong dual is barrelled. The distinguished property for spaces C_p(X) of continuous real-valued functions over a Tychonoff space X is a peculiar (although applicable) property. It is known that C_p(X) is distinguished if and only if C_p(X) is large in mathbb {R}^X if and only if X is a Delta -space (in sense of Reed) if and only if the strong dual of C_p(X) carries the finest locally convex topology. Our main results about spaces whose strong dual has only finite-dimensional bounded sets (see Theorems 2, 7 and Proposition 4) are used to study distinguished spaces C_k(X) with the compact-open topology. We also put together several known facts (Theorem 6) about distinguished spaces C_p(X) with self-contained full proofs.