We find sufficient conditions for a cotriad of which the objects are locally trivial fibrations, in order that the push-out be a locally trivial fibration. As an application, the universal $G$-bundle of a finite group $G$, and the classifying space is modeled by locally finite spaces. In particular, if $G$ is finite, then the universal $G$-bundle is the limit of an ascending chain of finite spaces. The bundle projection is a covering projection.

We prove that the third symmetric product of a chainable continuum has the fixed point property.

In his paper "Continuous mappings on continua" [5], T. Maćkowiak collected results concerning mappings on metric continua. These results are theorems, counterexamples, and unsolved problems and are listed in a series of tables at the ends of chapters. It is the purpose of the present paper to provide solutions (three proofs and one example) to four of those problems.

In un progetto di generalizzazione delle classiche topologie di tipo «set-open» di Arens-Dugundji introduciamo un metodo generale per produrre topologie in spazi di funzioni mediante l'uso di ipertopologie. Siano $X$, $Y$ spazi topologici e $C\left(X,Y\right)$ l'insieme delle funzioni continue da $X$ verso $Y$. Fissato un «network» $\alpha$ nel dominio $X$ ed una topologia $\tau$ nell'iperspazio $CL\left(Y\right)$ del codominio $Y$ si genera una topologia ${\tau }_{\alpha }$ in $C\left(X,Y\right)$ richiedendo che una rete $\left\{f_{\lambda }\right\}$ di $C\left(X,Y\right)$ converge in ${\tau }_{\alpha }$ ad $f\in C\left(X,Y\right)$ se e solo se la rete $\left\{\overline{f_{\lambda }\left(A\right)}\right\}$ converge in $\tau$ ad $\overline{f\left(A\right)}$...

Necessary conditions and sufficient conditions are given for $C_p\left(X\right)$ to be a (σ-) m₁- or m₃-space. (A space is an m₁-space if each of its points has a closure-preserving local base.) A compact uncountable space K is given with $C_p\left(K\right)$ an m₁-space, which answers questions raised by Dow, Ramírez Martínez and Tkachuk (2010) and Tkachuk (2011).

Given a space $X$, its $G_{\delta }$-subsets form a basis of a new space $X_{\omega }$, called the $G_{\delta }$-modification of $X$. We study how the assumption that the $G_{\delta }$-modification $X_{\omega }$ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_{\omega }$ is discrete and, hence, homogeneous. Thus, $X_{\omega }$ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_{\delta }$-modification of $X$ is homogeneous, then the weight $w\left(X\right)$ of $X$ does not exceed $2^{\omega }$ (Theorem 1). We also establish...

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where $A_x$ =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a ${\sum }_2^0$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [${\sum }_1^1\left(X^2\right)]={\sum }_1^1\left(X\right)$ for a wide class of ${\sum }_2^0$-supported σ-ideals.