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

A. M. Mohamad


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:
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:
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
end {enumerate}
end {enumerate}

quasi--regular--$G_{delta}$--diagonal; $wDelta$--space; developable; $3$--link property.

Math Review Classification
Primary 54E30, 54E35

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7 pages

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