We study a class of functions which differ essentially from those which are the sum of a convex function and a regular one and which have interesting properties related to $\mathrm{\Gamma }$-convergence and to problems with non-convex constraints. In particular some results are given for the associated evolution equations.

For a locally symmetric space $M$, we define a compactification $M\cup M\left(\infty \right)$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M\left(\infty \right)$ to certain geodesics in $M$. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give $M\left(\infty \right)$ for locally symmetric spaces. Moreover, $M\left(\infty \right)$ has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in...

In this paper we introduce and study the concepts of $error$-closed set and $error$-limit ($error$-cluster) points of $L$-nets and $L$-ideals using the notion of almost $N$-compact remoted neighbourhoods in $L$-topological spaces. Then we introduce and study the concept of $error$-continuous mappings. Several characterizations based on $error$-closed sets and the $error$-convergence theory of $L$-nets and $L$-ideals are presented for $error$-continuous mappings.

A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces $X$ which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel’skii ${\alpha }_1$ spaces, for which every sheaf at a point can be amalgamated in a natural way. Let $C_p\left(X\right)$ denote the space of continuous real-valued functions on $X$ with the topology of pointwise convergence. Our main result...

We extend the idea of $I$-convergence and $I^{*}$-convergence of sequences to a topological space and derive several basic properties of these concepts in the topological space.

In this paper we introduce the $ℐ$- and ${ℐ}^{*}$-convergence and divergence of nets in $\left(\ell \right)$-groups. We prove some theorems relating different types of convergence/divergence for nets in $\left(\ell \right)$-group setting, in relation with ideals. We consider both order and $\left(D\right)$-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that ${ℐ}^{*}$-convergence/divergence implies $ℐ$-convergence/divergence for every ideal, admissible for...

We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P....

This paper completes and improves results of [10]. Let $(X,d_{{}_X})$, $(Y,d_{{}_Y})$ be two metric spaces and $G$ be the space of all $Y$-valued continuous functions whose domain is a closed subset of $X$. If $X$ is a locally compact metric space, then the Kuratowski convergence ${\tau }_{{}_K}$ and the Kuratowski convergence on compacta ${\tau }_{{}_K}^c$ coincide on $G$. Thus if $X$ and $Y$ are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology ${\tau }_{{}_{AW}}$ (generated by the box metric of $d_{{}_X}$ and $d_{{}_Y}$) and ${\tau }_{{}_K}^c$ convergence on $G$,...

We investigate free groups over sequential spaces. In particular, we show that the free $k$-group and the free sequential group over a sequential space with unique limits coincide and, barred the trivial case, their sequential order is ${\omega }_1$.