In the present paper we introduce a convergence condition $\left({\Sigma }^{\text{'}}\right)$ and continue the study of “not distinguish” for various kinds of convergence of sequences of real functions on a topological space started in [2] and [3]. We compute cardinal invariants associated with introduced properties of spaces.

In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of $ℐ$-quasinormal convergence. We then introduce the notion of $ℐQN\left(ℐwQN\right)$ space as a topological space in which every sequence of continuous real valued functions pointwise converging to $0$, is also $ℐ$-quasinormally convergent to $0$ (has a subsequence which is $ℐ$-quasinormally convergent to $0$) and make certain observations on those spaces.

We show that if $X$ is first-countable, of countable extent, and a subspace of some ordinal, then $C_p(X,2)$ is Lindelöf.

A dense-in-itself space $X$ is called $C_{\square }$-discrete if the space of real continuous functions on $X$ with its box topology, $C_{\square }\left(X\right)$, is a discrete space. A space $X$ is called almost-$\omega$-resolvable provided that $X$ is the union of a countable increasing family of subsets each of them with an empty interior. We analyze these classes of spaces by determining their relations with $\kappa$-resolvable and almost resolvable spaces. We prove that every almost-$\omega$-resolvable space is $C_{\square }$-discrete, and that these classes coincide in...

For a Tychonoff space $X$, we will denote by $X_0$ the set of its isolated points and $X_1$ will be equal to $X\setminus X_0$. The symbol $C\left(X\right)$ denotes the space of real-valued continuous functions defined on $X$. $\square {ℝ}^{\kappa }$ is the Cartesian product ${ℝ}^{\kappa }$ with its box topology, and $C_{\square }\left(X\right)$ is $C\left(X\right)$ with the topology inherited from $\square {ℝ}^X$. By $\widehat{C}\left(X_1\right)$ we denote the set $\{f\in C\left(X_1\right):f$ can be continuously extended to all of $X\}$. A space $X$ is almost-$\omega$-resolvable if it can be partitioned by a countable family of subsets in such a way that every non-empty open subset of $X$ has a non-empty...

We investigate spaces $C_p(·,n)$ over LOTS (linearly ordered topological spaces). We find natural necessary conditions for linear Lindelöfness of $C_p(·,n)$ over LOTS. We also characterize countably compact LOTS whose $C_p(·,n)$ is linearly Lindelöf for each n. Both the necessary conditions and the characterization are given in terms of the topology of the Dedekind completion of a LOTS.

For a metrizable space X and a finite measure space (Ω, $𝔐$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $𝔐$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by $USC{C}_B\left(X\right)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $\phi \in USC{C}_B\left(X\right)$ with its graph which is a closed subset of X × ℝ. The space $USC{C}_B\left(X\right)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $USC{C}_B\left(X\right)$ is homeomorphic to a...

Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear measure then...

Let (X,f) be a dynamical system. In general the set of all ω-limit sets of f is not closed in the hyperspace of closed subsets of X. In this paper we study the case when X is a graph, and show that the family of ω-limit sets of a graph map is closed with respect to the Hausdorff metric.