The present paper aims to furnish simple proofs of some recent results about selections on product spaces obtained by García-Ferreira, Miyazaki and Nogura. The topic is discussed in the framework of a result of Katětov about complete normality of products. Also, some applications for products with a countably compact factor are demonstrated as well.

It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint $G_{\delta }$-sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

Let $\kappa$ be a cardinal number with the usual order topology. We prove that all subspaces of ${\kappa }^2$ are weakly sequentially complete and, as a corollary, all subspaces of ${\omega }_1^2$ are sequentially complete. Moreover we show that a subspace of ${({\omega }_1+1)}^2$ need not be sequentially complete, but note that $X=A\times B$ is sequentially complete whenever $A$ and $B$ are subspaces of $\kappa$.

Certain equivalences of Mrowka's separating condition enable us to characterize when parametric maps are open, closed or quotient.

We first provide a modified version of the proof in [3] that the Sorgenfrey line is T1. Here, we prove that it is in fact T2, a stronger result. Next, we prove that all subspaces of ℝ1 (that is the real line with the usual topology) are Lindel¨of. We utilize this result in the proof that the Sorgenfrey line is Lindel¨of, which is based on the proof found in [8]. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane...

Paracompactness ($=2$-paracompactness) and normality of a subspace $Y$ in a space $X$ defined by Arhangel’skii and Genedi [4] are fundamental in the study of relative topological properties ([2], [3]). These notions have been investigated by primary using of the notion of weak $C$- or weak $P$-embeddings, which are extension properties of functions defined in [2] or [18]. In fact, Bella and Yaschenko [8] characterized Tychonoff spaces which are normal in every larger Tychonoff space, and this result is essentially...

For a cardinal $\alpha$, we say that a subset $B$ of a space $X$ is $C_{\alpha }$-compact in $X$ if for every continuous function $fX\to {ℝ}^{\alpha }$, $f\left[B\right]$ is a compact subset of ${ℝ}^{\alpha }$. If $B$ is a $C$-compact subset of a space $X$, then $\rho (B,X)$ denotes the degree of $C_{\alpha }$-compactness of $B$ in $X$. A space $X$ is called $\alpha$-pseudocompact if $X$ is $C_{\alpha }$-compact into itself. For each cardinal $\alpha$, we give an example of an $\alpha$-pseudocompact space $X$ such that $X\times X$ is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

We present several results related to $L\Sigma \left(\kappa \right)$-spaces where $\kappa$ is a finite cardinal or $\omega$; we consider products and some constructions that lead from spaces of these classes to other spaces of similar classes.

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...

For a topological property $P$, we say that a space $X$ is star $P$ if for every open cover $𝒰$ of the space $X$ there exists $A\subset X$ such that $st(A,𝒰)=X$. We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf space with...