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A rotoid is a space X with a special point e ∈ X and a homeomorphism F: X² → X² having F(x,x) = (x,e) and F(e,x) = (e,x) for every x ∈ X. If any point of X can be used as the point e, then X is called a strong rotoid. We study some general properties of rotoids and prove that the Sorgenfrey line is a strong rotoid, thereby answering several questions posed by A. V. Arhangel'skii, and we pose further questions.

In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D\subset X\times X$ with $\left|D\right|=d\left(X\right)$. We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $\Sigma$-space $X$ and hence $X^{\omega }$ is $d$-separable. We give an example of a countably compact space $X$ such that $X^{\omega }$ is not $d$-separable. On the other hand, we show that for any Lindelöf $p$-space $X$ there exists a discrete subset $D\subset X\times X$ such that $\Delta =\{(x,x):x\in X\}\subset \overline{D}$; in particular, the diagonal $\Delta$ is a retract of $\overline{D}$ and the projection...

It is shown that certain weak-base structures on a topological space give a $D$-space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are $D$-spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Hence, quotient mappings, with compact fibers, from metric spaces have a $D$-space image. What about quotient $s$-mappings? Arhangel’skii and Buzyakova have shown that spaces with a point-countable base are $D$-spaces...