English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a Urysohn space $X$ we define the regular diagonal degree$\overline{\Delta}(X)$ of $X$ to be the minimal infinite cardinal $\kappa$ suchthat $X$ has a regular $G_\kappa$-diagonal i.e. there is a family$(U_\eta:\eta<\kappa)$ of open neighborhoods of $\Delta_X=\{(x,x)\in X^2:x\inX\}$ in $X^2$ such that $\Delta_X = \bigcap_{\eta<\kappa} \overline{U}_\eta$.  In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\leq2^{c(X)\cdot\overline{\Delta}(X)}$; (2) $|X|\leq 2^{\overline{\Delta}(X)\cdot2^{wL(X)}}$; (3) $|X|\le wL(X)^{\overline{\Delta}(X)\cdot\chi(X)}$; and (4)$|X|\le aL(X)^{\overline{\Delta}(X)}$; where $\chi(X)$, $c(X)$, $wL(X)$ and$aL(X)$ are respectively the character, the cellularity, the weak Lindel\"ofnumber and the almost Lindel\"of number of $X$.  The first inequality extends to the uncountable case Buzyakova's result thatthe cardinality of a ccc-space with a regular $G_\delta$-diagonal does notexceed $2^\omega$. It follows from (2) that every weakly Lindel\"of space witha regular $G_\delta$-diagonal has cardinality at most $2^{2^\omega}$.  Inequality (3) implies that when $X$ is a space with a regular$G_\delta$-diagonal then $|X|\le wL(X)^{\chi(X)}$. This improves significantlyBell, Ginsburg and Woods inequality $|X|\le 2^{\chi(X)wL(X)}$ for the class ofnormal spaces with regular $G_\delta$-diagonals. In particular (3) shows thatthe cardinality of every first countable space with a regular$G_\delta$-diagonal does not exceed $wL(X)^\omega$.  For the class of spaces with regular $G_\delta$-diagonals (4) improves Bellaand Cammaroto inequality $|X|\le 2^{\chi(X)\cdot aL(X)}$, which is valid forall Urysohn spaces. Also, it follows from (4) that the cardinality of everyspace with a regular $G_\delta$-diagonal does not exceed $aL(X)^\omega$.

Cardinalities of weakly Lindel\"of spaces with regular $G_\kappa$-diagonals