Reverse induction proof of D property of the countable power of the Sorgenfrey line

On the spread of topological groups containing subsets of the Sorgenfrey line

We prove that any topological group $G$ containing a subspace $X$ of the Sorgenfrey line has spread $s(G)\ge s(X\times X)$. Under OCA, each topological group containing an uncountable subspace of the Sorgenfrey line has uncountable spread. This implies that under OCA a cometrizable topological group $G$ is cosmic if and only if it has countable spread. On the other hand, under CH there exists a cometrizable Abelian topological group that has hereditarily Lindelöf countable power and contains an uncountable subspace of the Sorgenfrey line. This cometrizable topological group has countable sprea…

When the property of having a $π$-tree is preserved by products

We find sufficient conditions under which the product of spaces that have a $π$-tree also has a $π$-tree. These conditions give new examples of spaces with a $π$-tree: every at most countable power of the Sorgenfrey line and every at most countable power of the irrational Sorgenfrey line has a $π\!$-tree. Also we show that if a space has a $π$-tree, then its product with the Baire space, with the Sorgenfrey line, and with the countable power of the Sorgenfrey line also has a $π$-tree.

On $C$-embedded subspaces of the Sorgenfrey plane

We prove that every $C^*$-embedded subset of $ß$ is a hereditarily Baire subspace of $\mathbb R^2$. We also show that for a subspace $E\subseteq\{(x,-x):x\in\mathbb R\}$ of the Sorgenfrey plane $\mathbb S^2$ the following conditions are equivalent: (i) $E$ is $C$-embedded in $\mathbb S^2$; (ii) $E$ is $C^*$-embedded in $\mathbb S^2$; (iii) $E$ is a countable $G_δ$-subspace of $\mathbb R^2$ and (iv) $E$ is a countable functionally closed subspace of $\mathbb S^2$.

Sorgenfrey 线平方中的正常性

Normality in the square of the Sorgenfrey Line

研究实数集在Sorgenfrey下限拓扑下的正常性和伪正常性，探讨Q集和λ集的性质及在CH假设下的特殊情况。