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T_4 submetacompact space smaller than first measurable cardinal is realcompact #1212

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T_4 submetacompact space smaller than first measurable cardinal is realcompact #1212

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10 changes: 6 additions & 4 deletions theorems/T000382.md
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Expand Up @@ -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.
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