On Spaces with Quasi-Regular-$G_{delta}$-Diagonals

Abstract

This paper studies spaces with quasi--regular--$G_{delta}$--diagonal. It is shown that if $X$ is a normal space, then the following are equivalent: begin{enumerate} item $X$ admits a development satisfying the $3$--link property. item $X$ is a $wDelta$ with quasi--regular--$G_{delta}$--diagonal. item $X$ is a $wDelta$ with regular--$G_{delta}$--diagonal. item $X$ is $K$--semimetrizable via a semimetric satisfying $(AN)$. item There is a semimetric $d$ on $X$ such that: begin{enumerate} item [a.] if $langle x_n rangle$ and $langle y_n rangle$ are sequences both converging to the same point, then lim $d(x_n,y_n) = 0$, and item [b.] if $x$ and $y$ are distinct points of $X$ and $langle x_n rangle$ and $langle y_n rangle$ are sequences converging to $x$ and $y$, respectively, then there are integers $L$ and $M$ such that if $n > L$, then $d(x_n,y_n) > frac {1}{M}$. end {enumerate} end {enumerate}