Perfect images of generalized ordered spaces
Volume 240 / 2018
We study the class of perfect images of generalized ordered (GO) spaces, which we denote by PIGO. Mary Ellen Rudin’s celebrated result characterizing compact monotonically normal spaces as the continuous images of compact linearly ordered spaces implies that every space with a monotonically normal compactification is in PIGO. But PIGO is wider: every metrizable space is in the class, but not every metrizable space has a monotonically normal compactification. On the other hand, a locally compact space is in PIGO if and only if it has a monotonically normal compactification. We answer a question of Bennett and Lutzer whether a (semi)stratifiable space with a monotonically normal compactification must be metrizable by showing that any semistratifiable member of PIGO is metrizable. This also shows that there are monotonically normal spaces which are not in PIGO. We investigate cardinal functions in PIGO, and in particular show that if $K$ is a compact subset of a space $X$ in PIGO, then the character of $K$ in $X$ equals the pseudo-character of $K$ in $X$. We show that the product of two nondiscrete spaces in PIGO is not in PIGO unless both are metrizable or neither one contains a countable set with a limit point. Finally, we look at the narrower class of perfect images of linearly ordered spaces, which we denote by PILOTS. Every metrizable space is in PILOTS, and if a space in PILOTS has a $G_\delta $-diagonal, then it must be metrizable. Thus familiar GO-spaces such as the Sorgenfrey line and the Michael line are in PIGO but not in PILOTS.