We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c₀(Γ) for any set Γ. The first one is [0,ω₁] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω₀ + 1.

On cherche à donner une construction aussi simple que possible d'un borélien donné d'un produit de deux espaces polonais. D'où l'introduction de la notion de classe de Wadge potentielle. On étudie notamment ce que signifie "ne pas être potentiellement fermé", en montrant des résultats de type Hurewicz. Ceci nous amène naturellement à des théorèmes d'uniformisation partielle, sur des parties "grosses", au sens du cardinal ou de la catégorie.

The Kuratowski-Dugundji theorem that a metrizable space is an absolute (neighborhood) extensor in dimension n iff it is $L{C}^{n-1}{C}^{n-1}$ (resp., $L{C}^{n-1}$) is extended to a class of non-metrizable absolute (neighborhood) extensors in dimension n. On this base, several facts concerning metrizable extensors are established for non-metrizable ones.

We conclude the classification of spaces of continuous functions on ordinals carried out by Górak [Górak R., Function spaces on ordinals, Comment. Math. Univ. Carolin. 46 (2005), no. 1, 93–103]. This gives a complete topological classification of the spaces $C_p\left([0,\alpha ]\right)$ of all continuous real-valued functions on compact segments of ordinals endowed with the topology of pointwise convergence. Moreover, this topological classification of the spaces $C_p\left([0,\alpha ]\right)$ completely coincides with their uniform classification.

Kechris and Louveau in [5] classified the bounded Baire-1 functions, which are defined on a compact metric space $K$, to the subclasses ${ℬ}_1^{\xi }\left(K\right)$, $\xi <{\omega }_1$. In [8], for every ordinal $\xi <{\omega }_1$ we define a new type of convergence for sequences of real-valued functions ($\xi$-uniformly pointwise) which is between uniform and pointwise convergence. In this paper using this type of convergence we obtain a classification of pointwise convergent sequences of continuous real-valued functions defined on a compact metric space $K$, and...

The main result of the present paper is a classification theorem for finite-sheeted covering mappings over connected paracompact spaces. This theorem is a generalization of the classical classification theorem for covering mappings over a connected locally pathwise connected semi-locally 1-connected space in the finite-sheeted case. To achieve the result we use the classification theorem for overlay structures which was recently proved by S. Mardesic and V. Matijevic (Theorems 1 and 4 of [5]).

For every metric space X we introduce two cardinal characteristics $co{v}^{♭}\left(X\right)$ and $co{v}^{♯}\left(X\right)$ describing the capacity of balls in X. We prove that these cardinal characteristics are invariant under coarse equivalence, and that two ultrametric spaces X,Y are coarsely equivalent if $co{v}^{♭}\left(X\right)=co{v}^{♯}\left(X\right)=co{v}^{♭}\left(Y\right)=co{v}^{♯}\left(Y\right)$. This implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $co{v}^{♭}\left(X\right)=co{v}^{♯}\left(X\right)$. Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and only if $co{v}^{♯}\left(X\right)=co{v}^{♯}\left(Y\right)$...

Some results on cleavability theory are presented. We also show some new [16]'s results.

Clone properties are the properties expressible by the first order sentence of the clone language. The present paper is a contribution to the field of problems asking when distinct sentences of the language determine distinct topological properties. We fully clarify the relations among the rigidity, the fix-point property, the image-determining property and the coconnectedness.