arXiv subjects

Ahmed Bouziad

Publications and source records attributed to Ahmed Bouziad.

7 recordsLinked to original sources

arXiv2022

A note on the Schur and Phillips lemmas

It is well-known that every weakly convergent sequence in $\ell_1$ is convergent in the norm topology (Schur's lemma). Phillips' lemma asserts even more strongly that if a sequence $(μ_n)_{n\in\mathbb N}$ in $\ell_\infty'$ converges pointwise on $\{0,1\}^\mathbb N$ to $0$, then its $\ell_1$-projection converges in norm to $0$. In this note we show how the second category version of Schur's lemma, for which a short proof is included, can be used to replace in Phillips' lemma $\{0,1\}^\mathbb N$ by any of its subsets which contains all finite sets and having some kind of interpolation property for finite sets.

math.FA↗

arXiv2019

On proximal fineness of topological groups in their right uniformity

A uniform space $X$ is said to be proximally fine if every proximally continuous map on $X$ into a uniform is uniformly continuous. We supply a proof that every topological group which is functionnaly generated by its precompact subsets is proximally fine with respect to its right uniformity. On the other hand, we show that there are various permutation groups $G$ on the integers $\mathbb N$ that are not proximally fine with respect to the topology generated by the sets $\{g\in G: g(A)\subset B\}$, $A,B\subset \mathbb N$.

math.GN↗

arXiv2019

Preservation of uniform continuity under pointwise product

Let $X$ be a uniform space and $U(X)$ the linear space of real-valued uniformly continuous functions on $X$. Our main objective is to give a number of properties characterizing the fact that $U(X)$ is stable under pointwise product in case $X$ is a metric space. Some of these characterizations hold in much more general circumstances.

math.GN↗

arXiv2010

Cliquishness and Quasicontinuity of Two Variables Maps

We study the existence of continuity points for mappings $f: X\times Y\to Z$ whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are fragmentable and $y$-sections $X\ni x\to f(x,y)\in Z$ are quasicontinuous, where $X$ is a Baire space and $Z$ is a metric space. For the factor $Y$, we consider two infinite "point-picking" games $G_1(y)$ and $G_2(y)$ defined respectively for each $y\in Y$ as follows: In the $n$th inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$, then Player II picks a point $y_n\inD_n$; II wins if $y$ is in the closure of $\{y_n:n\in\mathbb N\}$, otherwise I wins. It is shown that (i) $f$ is cliquish if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $f$ is quasicontinuous if the $x$-sections of $f$ are continuous and the set of $y\in Y$ such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially a result of Debs (1986) and item (ii) indicates that the problem of Talagrand (1985) on separately continuous maps has a positive answer for a wide class of "small" compact spaces.

math.GN↗

arXiv2010

Lower Quasicontinuity, Joint Continuity and Related concepts

Let $X$ and $Y$ be topological spaces, let $Z$ be a metric space, and let $f: X\times Y\to Z$ be a mapping. It is shown that when $Y$ has a countable base $\mathcal B$, then under a rather general condition on the set-valued mappings $X\ni x\to f_x(B)\in 2^Z$, $B\in\mathcal B$, there is a residual set $R\subset X$ such that for every $(a,b)\in R\times Y$, $f$ is jointly continuous at $(a,b)$ if (and only if) $f_a: Y\to Z$ is continuous at $b$. Several new results are also established when the notion of continuity is replaced by that of quasicontinuity or by that of cliquishness. Our approach allows us to unify and improve various results from the literature.

math.GN↗

arXiv2010

Itzkowitz's problem for group of finite exponent

Itzkowitz's problem asks whether every topological group $G$ has equal left and right uniform structures provided that bounded left uniformly continuous real-valued function on $G$ are right uniformly continuous. This paper provides a positive answer to this problem if $G$ is of bounded exponent or, more generally, if there exist an integer $p\geq 2$ and a nonempty open set $U\subset G$ such that the power map $U\ni g\to g^p\in G$ is left (or right) uniformly continuous. This also resolves the problem for periodic groups which are Baire spaces.

math.GR↗

arXiv2009

Some results on separate and joint continuity

Let $f: X\times K\to \mathbb R$ be a separately continuous function and $\mathcal C$ a countable collection of subsets of $K$. Following a result of Calbrix and Troallic, there is a residual set of points $x\in X$ such that $f$ is jointly continuous at each point of $\{x\}\times Q$, where $Q$ is the set of $y\in K$ for which the collection $\mathcal C$ includes a basis of neighborhoods in $K$. The particular case when the factor $K$ is second countable was recently extended by Moors and Kenderov to any Čech-complete Lindelöf space $K$ and Lindelöf $α$-favorable $X$, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when $K$ is a Lindelöf $p$-space and $X$ is conditionally $σ$-$α$-favorable space. Here we add new results of this sort when the factor $X$ is $σ_{C(X)}$-$β$-defavorable and when the assumption "base of neighborhoods" in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.

math.GN↗