Abstract:
In this paper we show that a quasi--$G^*_{delta}$--diagonal plays a central role in metrizability. We prove that: if $X$ is a first--countable $GO$--space, then $X$ is metrizable if and only if $X$ is quasi--$sigma$--space; a $wtheta$--space is metrizable if and only if it is a quasi--Nagata space with a quasi--$G^*_{delta}(2)$--diagonal; a linearly ordered space $X$ with a quasi--$G^*_{delta}(2)$--diagonal is a $Theta$--space; a space $X$ is developable if and only if it is a $wtheta$, $beta$--space with a quasi--$G^*_{delta}(2)$--diagonal.