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We say that a space $X$ has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. A space $X$ has a zeroset diagonal if there is a continuous mapping $f:X^2\to [0,1]$ with ${\Delta }_X=f^{-1}\left(0\right)$, where ${\Delta }_X=\{(x,x):x\in X\}$. In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most $𝔠$.

A topological space $X$ is said to be star Lindelöf if for any open cover $𝒰$ of $X$ there is a Lindelöf subspace $A\subset X$ such that $St(A,𝒰)=X$. The “extent” $e\left(X\right)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$. We prove that under $V=L$ every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under $\mathrm{MA}+\neg \mathrm{CH}$, which shows that a star Lindelöf, first countable and normal space may not have countable extent.

We prove that, assuming , if $X$ is a space with ${\aleph }_1$-calibre and a zeroset diagonal, then $X$ is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular $G_{\delta }$-diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with ${\aleph }_1$-calibre.

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If $X$ is a semi-stratifiable space, then $X$ is separable if and only if $X$ is $DC\left({\omega }_1\right)$; (2) If $X$ is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then $X$ is separable; (3) Let $X$ be a $\omega$-monolithic star countable extent semi-stratifiable space. If $t\left(X\right)=\omega$ and $d\left(X\right)\le {\omega }_1$, then $X$ is hereditarily separable. Finally, we prove that for any $T_1$-space...

A topological space $X$ has a rank 2-diagonal if there exists a diagonal sequence on $X$ of rank $2$, that is, there is a countable family $\{{𝒰}_n:n\in \omega \}$ of open covers of $X$ such that for each $x\in X$, $\left\{x\right\}=\bigcap \{{\mathrm{St}}^2(x,{𝒰}_n):n\in \omega \}$. We say that a space $X$ satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. We mainly prove that if $X$ is a DCCC normal space with a rank 2-diagonal, then the cardinality of $X$ is at most $𝔠$. Moreover, we prove that if $X$ is a first countable...