### $\aleph $-Dowker spaces

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We construct a space having the properties in the title, and with the same technique, a countably compact ${T}_{2}$ topological group which is not absolutely countably compact.

We exhibit a metric continuum X and a polyhedron P such that the Cartesian product X × P fails to be the product of X and P in the shape category of topological spaces.

We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, $\omega $-stable and $\omega $-monolithic. It is also established that any Sokolov compact space $X$ is Fréchet-Urysohn and the space ${C}_{p}\left(X\right)$ is Lindelöf. We prove that any Sokolov space with a ${G}_{\delta}$-diagonal has a countable network and obtain some cardinality restrictions on subsets...

Let $P$ be a topological property. A space $X$ is said to be star P if whenever $\mathcal{U}$ is an open cover of $X$, there exists a subspace $A\subseteq X$ with property $P$ such that $X=St(A,\mathcal{U})$. In this note, we construct a Tychonoff pseudocompact SCE-space which is not star Lindelöf, which gives a negative answer to a question of Rojas-Sánchez and Tamariz-Mascarúa.

We observe the existence of a $\sigma $-compact, separable topological group $G$ and a countable topological group $H$ such that the tightness of $G$ is countable, but the tightness of $G\times H$ is equal to $\U0001d520$.