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 $𝐑$.

In this article, we investigate new topological descriptions for two well-known mappings ${𝒵}_A$ and ${\Im }_A$ defined on intermediate rings $A\left(X\right)$ of $C\left(X\right)$. Using this, coincidence of each two classes of $z$-ideals, ${𝒵}_A$-ideals and ${\Im }_A$-ideals of $A\left(X\right)$ is studied. Moreover, we answer five questions concerning the mapping ${\Im }_A$ raised in [J. Sack, S. Watson, $C$ and $C^{*}$ among intermediate rings, Topology Proc. 43 (2014), 69–82].

In this paper we deal with the maximal monotonicity of A + B when the two maximal monotone operators A and B defined in a Hilbert space X are satisfying the condition: Uλ ≥ 0 λ (dom B - dom A) is a closed linear subspace of X.

We present J. Jezierski's approach to the Nielsen fixed point theory for a broad class of multivalued mappings [Je1]. We also describe some generalizations and different techniques existing in the literature.

We extend a theorem of S. Claytor in order to characterize the Peano generalized continua which are embeddable into the 2-sphere. We also give a characterization of the Peano generalized continua which admit closed embeddings in the Euclidean plane.

The characterization of the pointwise limits of the sequences of Świątkowski functions is given. Modifications of Świątkowski property with respect to different topologies finer than the Euclidean topology are discussed.

We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous...