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.