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Perfect images of generalized ordered spaces

Volume 240 / 2018

Gary Gruenhage, David J. Lutzer Fundamenta Mathematicae 240 (2018), 175-197 MSC: Primary 54F05; Secondary 54D35, 54C10, 54E20, 54E35, 54D15. DOI: 10.4064/fm343-1-2017 Published online: 23 June 2017

Abstract

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.

Authors

  • Gary GruenhageDepartment of Mathematics and Statistics
    Auburn University
    Auburn, AL 36849, U.S.A.
    e-mail
  • David J. LutzerDepartment of Mathematics
    College of William and Mary
    Williamsburg, VA 23187, U.S.A.
    e-mail

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