For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have...

Let $M$ and $K$ be closed subsets of [0,1] with $K$ a subset of the limit points of $M$. Necessary and sufficient conditions are found for the existence of a continuous function $f:[0,1]\to [0,1]$ such that $M$ is an $\omega$-limit set for $f$ and $K$ is the set of fixed points of $f$ in $M$.

We introduce a functor of order-preserving functionals which contains some known functors as subfunctors. It is shown that this functor is weakly normal and generates a monad.

We prove that for n > 1 the space of proper maps P 0(n, k) and the space of local maps F 0(n, k) are not homotopy equivalent.

For any Borel ideal ℐ we describe the ℐ-Baire system generated by the family of quasi-continuous real-valued functions. We characterize the Borel ideals ℐ for which the ideal and ordinary Baire systems coincide.

The main goal of this paper is to characterize the family of all functions f which satisfy the following condition: whenever g is a Darboux function and f < g on ℝ there is a Darboux function h such that f < h < g on ℝ.

We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature $K$, we give a description of the totally geodesic unit vector fields for $K=0$ and $K=1$ and prove a non-existence result for $K\ne 0,1$. We also found a family ${\xi }_{\omega }$ of vector fields on the hyperbolic 2-plane $L^2$ of curvature $-c^2$ which generate foliations on $T_1{L}^2$ with leaves of constant intrinsic...

We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if $C_p\left(X\right)$ is Lindelöf, $Y=X\cup \left\{p\right\}$, and the point $p$ has countable character in $Y$, then $C_p\left(Y\right)$ is Lindelöf; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l\left(C_p{\left(Y\right)}^{\omega }\right)\le l\left(C_p{\left(X\right)}^{\omega }\right)$ and $ext\left(C_p{\left(Y\right)}^{\omega }\right)\le ext\left(C_p{\left(X\right)}^{\omega }\right)$.

Si dimostra che il funzionale ${\int }_{\mathrm{\Omega }}f(u,Du)dx$ è semicontinuo inferiormente su $W_{loc}^{1,1}(\mathrm{\Omega })$, rispetto alla topologia indotta da $L_{loc}^1(\mathrm{\Omega })$, qualora l’integrando $f(s,p)$ sia una funzione non-negativa, misurabile in $s$, convessa in $p$, limitata nell’intorno dei punti del tipo $(s,0)$, e tale che la funzione $s\mapsto f(s,0)$ sia semicontinua inferiormente su $𝐑$.