From 3c2dea3089692ebdde2dece112e60de563a88696 Mon Sep 17 00:00:00 2001 From: Moniker1998 Date: Sat, 25 Jan 2025 12:59:55 +0100 Subject: [PATCH] edited theorem to a stronger one --- theorems/T000382.md | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/theorems/T000382.md b/theorems/T000382.md index d783a19699..60496fe538 100644 --- a/theorems/T000382.md +++ b/theorems/T000382.md @@ -3,13 +3,15 @@ uid: T000382 if: and: - P000007: true - - P000031: true + - P000194: true - P000164: true then: P000162: true refs: -- doi: 10.4153/CJM-1972-081-9 - name: Certain Subsets of Products of Metacompact Spaces and Subparacompact Spaces are Realcompact + - doi: 10.1090/S0002-9939-1973-0322812-9 + name: Certain Subsets of Products of θ-refinable Spaces are Realcompact + - doi: 10.1007/978-1-4615-7819-2 + name: Rings of Continuous Functions (Gillman & Jerison) --- -See {{doi:10.4153/CJM-1972-081-9}} corollary 2. +By theorem in the article {{doi:10.1090/S0002-9939-1973-0322812-9}}, if $X$ is $T_4$, embeds as a closed subspace of product of $\theta$-refinable spaces, and every closed discrete subspace of $X$ is realcompact (equivalently, of cardinality smaller than the first measurable cardinal, see {{doi:10.1007/978-1-4615-7819-2}} theorem 12.2), then $X$ is realcompact. If $X$ is already $\theta$-refinable, then it's itself a product of $\theta$-refinable spaces in which it embeds as a closed subspace.