In this paper, on the family (Y) of all open subsets of a space Y we define the so called quasi Scott topology, denoted by ${\tau }_{qSc}$. This topology defines in a standard way, on the set C(Y,Z) of all continuous maps of the space Y to a space Z, a topology $t_{qIs}$ called the quasi Isbell topology. The latter topology is always larger than or equal to the Isbell topology, and smaller than or equal to the strong Isbell topology. Results and problems concerning the topology $t_{qIs}$ are given.

Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the $m$-topology on $C\left(X\right)$, denoted $C_m\left(X\right)$, and demonstrated that certain topological properties of $X$ could be characterized by certain topological properties of $C_m\left(X\right)$. For example, he showed that $X$ is pseudocompact if and only if $C_m\left(X\right)$ is a metrizable space; in this case the $m$-topology is precisely the topology of uniform convergence. What is interesting with regards to the $m$-topology is that it is possible, with...

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

The hyperspaces $ANR\left({ℝ}^n\right)$ and $AR\left({ℝ}^n\right)$ in $2^{{ℝ}^n}\left(n\ge 3\right)$ consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute $G_{\delta \sigma \delta }$-spaces and that, indeed, they are not $F_{\sigma \delta \sigma }$-spaces. The main result is that $ANR\left({ℝ}^n\right)$ is an absorber for the class of all absolute $G_{\delta \sigma \delta }$-spaces and is therefore homeomorphic to the standard model space ${\Omega }_3$ of this class.

Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $\varrho (x,y)=W\left(A_{xy}\right)$ where $A_{x,y}$ is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = y ∈ X:ϱ ( x,y) ≤ r coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value...

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.

By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight ≤ Sln(X) and...

In this paper we continue with the study of paracompact maps introduced in [1]. We give two external characterizations for paracompact maps including a characterization analogous to The Tamano Theorem in the category $𝒯OP$ (of topological spaces and continuous maps as morphisms). A necessary and sufficient condition for the Tychonoff product of a closed map and a compact map to be closed is also given.