From 3190ebf6db279a3b0cb3efe0fdd5e3d824a71da9 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Tue, 8 Oct 2024 10:24:07 -0400 Subject: [PATCH] Perfect space as alias for P132 (#790) --- properties/P000132.md | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/properties/P000132.md b/properties/P000132.md index d5dab9eae..5290f3e9f 100644 --- a/properties/P000132.md +++ b/properties/P000132.md @@ -1,16 +1,20 @@ --- uid: P000132 name: $G_\delta$ space +aliases: + - Perfect refs: - doi: 10.1007/978-1-4612-6290-9 name: Counterexamples in Topology - wikipedia: Gδ_space name: G-delta space + - zb: "0684.54001" + name: General Topology (Engelking, 1989) --- -A space in which every closed set is a $G_\delta$ (a countable intersection of open sets). -Equivalently, a space in which every open set is an $F_\sigma$ (a countable union of closed sets). +A space in which every closed set is a $G_\delta$ set (a countable intersection of open sets). +Equivalently, a space in which every open set is an $F_\sigma$ set (a countable union of closed sets). Defined on page 162 of {{doi:10.1007/978-1-4612-6290-9}}. -Note: A $G_\delta$ space is sometimes called a "perfect space" (Exercise 1.5.H(a) in {{zb:0684.54001}}). But that could be confused with a space that is a "perfect" in the sense of "perfect set" (= a set equal to its derived set), that is, a space without isolated point. See the discussion in . +Note: A $G_\delta$ space is sometimes called a "perfect space" (Exercise 1.5.H(a) in {{zb:0684.54001}}). Not to be be confused with a space that is a "perfect" in the sense of "perfect set" (= a set equal to its derived set = a closed set that is dense-in-itself), that is, a space without isolated point. See the discussion in .