Add Ellentuck topology S000223. (#1704)

Author: JSMassmann

spaces/S000223/README.md

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+---
+uid: S000223
+name: Ellentuck topology on $[\omega]^\omega$
+refs:
+  - zb: "0292.02054"
+    name: A new proof that analytic sets are Ramsey (E. Ellentuck)
+  - zb: "1007.03002"
+    name: Set theory (T. Jech, 2003)
+  - zb: "1400.03002"
+    name: Combinatorial set theory (L. Halbeisen, 2017)
+---
+
+Let $X = [\omega]^\omega$, the set of infinite sets of nonnegative integers, and give it the topology with basis consisting of all sets
+
+$$[s, A] = \{B \in X: s \subseteq B \subseteq s \cup A \;\text{ and }\; \max(s) < \min(B \setminus s)\}$$
+
+for a finite $s\subseteq\omega$ and an infinite $A\subseteq\omega$.
+If $s = \emptyset$ we let $\max(s) = -1$.
+
+*Note*: Omitting the condition $\max(s) < \min(B \setminus s)$ would give another base for the same topology, but the form chosen above is usually more convenient.
+
+Introduced by Ellentuck in {{zb:0292.02054}} (<https://www.jstor.org/stable/2272356>). See also Definition 26.25 in {{zb:1007.03002}} and the section "The Ellentuck Topology" on p. 248 of {{zb:1400.03002}}

spaces/S000223/properties/P000013.md

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+---
+space: S000223
+property: P000013
+value: false
+refs:
+  - zb: "0632.04005"
+    name: On completely Ramsey sets (S. Plewik)
+---
+
+See the remark after Proposition 4 of {{zb:0632.04005}}.
+$X$ is shown to be non-normal by applying Jones' lemma ({T836}) to a certain closed subspace $F\subseteq X$.

spaces/S000223/properties/P000028.md

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+---
+space: S000223
+property: P000028
+value: true
+---
+
+$\{[x \cap n, x]: n < \omega\}$ is a countable local base around $x$, for any $x \in [\omega]^\omega$: if $s, A$ are so that $x \in [s, A]$, that is $s \subseteq x \subseteq A \cup s$ and $\max(s) < \min(x \setminus s)$, then $s = x \cap (\max(s) + 1)$ and $[s, x] \subseteq [s, A]$. 

spaces/S000223/properties/P000029.md

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+---
+space: S000223
+property: P000029
+value: false
+---
+
+For $x: \omega \to 2$, let $A_x = \left\{\sum_{n = 0}^m 2^n x(n): m < \omega\right\}$. Then, when $x \neq y$, $A_x \cap A_y$ is finite, and so $\{[\emptyset, A_x]: x \in 2^\omega\}$ is an uncountable family of pairwise disjoint nonempty open sets.

spaces/S000223/properties/P000050.md

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+---
+space: S000223
+property: P000050
+value: true
+---
+
+Each basic open set $[s, A]$ with $s$ finite and $A$ infinite is clopen.
+Indeed, suppose $x\notin[s,A]$.
+If $x\cap[0,\max(s)]\ne s$, let $t=x\cap[0,\max(s)]$.
+Otherwise, necessarily $s\subseteq x$ and $\max(s)<\min(x\setminus s)$, and therefore $x\not\subseteq(s\cup A)$, in which case let $t=x\cap[0,\min(x\setminus(s\cup A)]$.
+In both cases, $x \in [t, x]$ and $[t, x] \cap [s, A] = \emptyset$.

spaces/S000223/properties/P000064.md

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+---
+space: S000223
+property: P000064
+value: true
+refs:
+  - zb: "0292.02054"
+    name: A new proof that analytic sets are Ramsey (E. Ellentuck)
+  - zb: "1007.03002"
+    name: Set theory (T. Jech, 2003)
+---
+
+Every meager set in $X$ is nowhere dense, which is a strong form of being Baire.
+See Corollary 8 in {{zb:0292.02054}} (<https://www.jstor.org/stable/2272356>) or Lemma 26.27(ii) in {{zb:1007.03002}}.

spaces/S000223/properties/P000065.md

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+---
+space: S000223
+property: P000065
+value: true
+---
+
+By construction.

spaces/S000223/properties/P000093.md

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+---
+space: S000223
+property: P000093
+value: false
+---
+
+Every non-empty open set has cardinality $\mathfrak{c}$ by construction.

spaces/S000223/properties/P000139.md

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+---
+space: S000223
+property: P000139
+value: false
+---
+
+Every nonempty open set has cardinality $\mathfrak{c}$.

spaces/S000223/properties/P000166.md

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+---
+space: S000223
+property: P000166
+value: true
+---
+
+When one only considers open sets $[s, A]$ where $A = \omega$, i.e. $\{x \in [\omega]^\omega: s \subseteq x \land \max(s) < \min(x \setminus s)\}$, then, the resulting topology is strictly coarser than the Ellentuck topology and homeomorphic to {S28}. Namely, conceiving {S28} as the set of infinite sequences of natural numbers, the map sending $\{a_n: n < \omega\}$, where $a_n < a_m$ for $n < m$, to $f: \omega \to \omega$ given by $f(0) = a_0$ and $f(n+1) = a_{n+1} - (a_n + 1)$, is a homeomorphism.