Add realcompactification of Rudin's space and basic properties

properties/P000008.md

@@ -13,3 +13,8 @@ A space which is both {P14} and {P3}.
 Equivalently, a space that is both {P14} and {P2}.
 
 Defined on page 12 of {{zb:0386.54001}} as "completely normal".
+
+----
+#### Meta-properties
+
+- This property is hereditary.

spaces/S000208/README.md

@@ -0,0 +1,16 @@
+---
+uid: S000208
+name: Hewitt realcompactification of Rudin's Dowker space
+refs:
+  - zb: "0224.54019"
+    name: A normal space X for which X×I is not normal (M.E. Rudin)
+  - doi: 10.1007/978-1-4615-7819-2
+    name: Rings of Continuous Functions (Gillman & Jerison)
+  - wikipedia: Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets
+    name: Cofinality on Wikipedia
+---
+
+$X$ is the subspace of the product $\prod_{n\in\omega}(\omega_{n+1}+1)$ with the box topology consisting of all $f\in \prod_{n\in \omega}(\omega_{n+1}+1)$ such that $\omega< \text{cf}(f(n))$ for all $n$ (see {{wikipedia:Cofinality#Cofinality_of_ordinals_and_other_well-ordered_sets}}).
+
+Defined (as the space called $X'$) and shown to be the Hewitt realcompactification of {S138} in section IV.4 of {{zb:0224.54019}} 
+(see remark 8.8 of {{doi:10.1007/978-1-4615-7819-2}} for the definition of Hewitt realcompactification).

spaces/S000208/properties/P000003.md

@@ -0,0 +1,7 @@
+---
+space: S000208
+property: P000003
+value: true
+---
+
+$X$ is a subspace of $\prod_n (\omega_{n+1}+1)$ with box topology, which is {P3}.

spaces/S000208/properties/P000008.md

@@ -0,0 +1,7 @@
+---
+space: S000208
+property: P000008
+value: false
+---
+
+$X$ contains {S138} and {S138|P8}.

spaces/S000208/properties/P000114.md

@@ -0,0 +1,7 @@
+---
+space: S000208
+property: P000114
+value: false
+---
+
+$X$ contains {S138}, and {S138|P114} and {S138|P57}.

spaces/S000208/properties/P000146.md

@@ -0,0 +1,12 @@
+---
+space: S000208
+property: P000146
+value: true
+refs:
+- zb: "0224.54019"
+  name: A normal space X for which X×I is not normal (M.E. Rudin)
+---
+
+See section IV.4 of {{zb:0224.54019}}.
+
+See also [K.P. Hart's notes](https://fa.ewi.tudelft.nl/~hart/37/onderwijs/old-courses/settop/rudin.pdf).

spaces/S000208/properties/P000147.md

@@ -0,0 +1,10 @@
+---
+space: S000208
+property: P000147
+value: true
+refs:
+- zb: "0224.54019"
+  name: A normal space X for which X×I is not normal (M.E. Rudin)
+---
+
+The proof is the same as for {S138}. See Lemma 4 in {{zb:0224.54019}}.

spaces/S000208/properties/P000164.md

@@ -0,0 +1,7 @@
+---
+space: S000208
+property: P000164
+value: true
+---
+
+$|X| \leq \aleph_\omega^\omega$ and $\aleph_\omega^\omega$ is smaller than every measurable cardinal.