We consider the following problem: Characterize the pairs ⟨A,B⟩ of subsets of ℝ which can be separated by a function from a given class, i.e., for which there exists a function f from that class such that f = 0 on A and f = 1 on B (the classical separation property) or f < 0 on A and f > 0 on B (a new separation property).

I discuss the number of iterations of the elementary sequential closure operation required to achieve the full sequential closure of a set in spaces of the form $C_p\left(X\right)$.

We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected $G_{\delta }$ graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected $G_{\delta }$ graph.

Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and $\alpha <{\omega }_1$ then f ○ g ∈ $B_{\alpha }\left(Z\right)$ for every $g\in B_{\alpha }\left(Z\right){\cap }^{Z}I$ if and only if f is continuous on I, where $B_{\alpha }\left(Z\right)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes $S_{\alpha }\left(Z\right)(\alpha >0)$ in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz...

Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space. We confirm...

Some properties of the Hausdorff distance in complete metric spaces are discussed. Results obtained in this paper explain ideas used in the theory of measures of noncompactness.

Let X be a completely regular Hausdorff topological space and $C_p\left(X\right)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_p\left(X\right)$, if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_p\left(X\right)$.

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.