Bien que les espaces de Berkovich définis sur un corps trop gros ne soient, en général, pas métrisables, nous montrons que leur topologie reste en grande partie gouvernée par les suites : tout point adhérent à une partie est limite d’une suite de points de cette partie et les parties compactes sont séquentiellement compactes. Notre preuve utilise de façon essentielle l’extension des scalaires et nous en étudions certaines propriétés. Nous montrons qu’un point d’un disque peut être défini sur un...

Let $A={\left(a_{n,k}\right)}_{n,k\ge 1}$ be a non-negative matrix. Denote by $L_{v,p,q,F}\left(A\right)$ the supremum of those $L$ that satisfy the inequality $${\parallel Ax\parallel }_{v,q,F}\ge L{\parallel x\parallel }_{v,p,F},$$ where $x\ge 0$ and $x\in l_p(v,F)$ and also $v={\left(v_n\right)}_{n=1}^{\infty }$ is an increasing, non-negative sequence of real numbers. If $p=q$, we use $L_{v,p,F}\left(A\right)$ instead of $L_{v,p,p,F}\left(A\right)$. In this paper we obtain a Hardy type formula for $L_{v,p,q,F}\left(H_{\mu }\right)$, where $H_{\mu }$ is a Hausdorff matrix and $0<q\le p\le 1$. Another purpose of this paper is to establish a lower bound for $\parallel A_W^{NM}{\parallel }_{v,p,F}$, where $A_W^{NM}$ is the Nörlund matrix associated with the sequence $W={\left\{w_n\right\}}_{n=1}^{\infty }$ and $1<p<\infty$. Our results generalize some works of Bennett, Jameson and present authors....

In this paper $ss$-quotient maps and $ssq$-spaces are introduced. It is shown that (1) countable tightness is characterized by $ss$-quotient maps and quotient maps; (2) a space has countable tightness if and only if it is a countably bi-quotient image of a locally countable space, which gives an answer for a question posed by F. Siwiec in 1975; (3) $ssq$-spaces are characterized as the $ss$-quotient images of metric spaces; (4) assuming $2^{\omega }<2^{{\omega }_1}$, a compact $T_2$-space is an $ssq$-space if and only if every countably compact subset...

Strongly sequential spaces were introduced and studied to solve a problem of Tanaka concerning the product of sequential topologies. In this paper, further properties of strongly sequential spaces are investigated.

It is proved that the product of two pseudo radial compact spaces is pseudo radial provided that one of them is monolithic.

Assuming OCA, we shall prove that for some pairs of Fréchet ${\alpha }_4$-spaces $X,Y$, the Fréchetness of the product $X\times Y$ implies that $X\times Y$ is ${\alpha }_4$. Assuming MA, we shall construct a pair of spaces satisfying the assumptions of the theorem.

We introduce the notion of a strongly Whyburn space, and show that a space $X$ is strongly Whyburn if and only if $X\times (\omega +1)$ is Whyburn. We also show that if $X\times Y$ is Whyburn for any Whyburn space $Y$, then $X$ is discrete.

A sufficient condition for the pseudo radiality of the product of two compact Hausdorff spaces is given.

In this paper, we give an affirmative answer to the problem posed by Y. Tanaka and Y. Ge (2006) in "Around quotient compact images of metric spaces, and symmetric spaces", Houston J. Math. 32 (2006) no. 1, 99-117.

The definitions of AP and WAP were originated in categorical topology by A. Pultr and A. Tozzi, Equationally closed subframes and representation of quotient spaces, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), no. 3, 167-183. In general, we have the implications: $T_2\Rightarrow KC\Rightarrow US\Rightarrow T_1$, where $KC$ is defined as the property that every compact subset is closed and $US$ is defined as the property that every convergent sequence has at most one limit. And a space is called submaximal if every dense subset is open....