Add Ellentuck topology S000223.

spaces/S000223/README.md

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+---
+uid: S000223
+name: Ellentuck topology on $[\omega]^\omega$
+refs:
+  - zb: "0546.03029"
+    name: Random isols (E. Ellentuck)
+---
+
+Let $X = [\omega]^\omega$, the set of infinite sets of natural numbers, and give it the basis consisting of $[s, A] = \{B \in X: s \subseteq B \subseteq A \cup s \land \max(s) < \min(B \setminus s)\}$ for finite and infinite, respectively, $s, A \subseteq \omega$. If $s = \emptyset$ we let $\max(s) = 0$.

spaces/S000223/properties/P000013.md

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+---
+space: S000223
+property: P000013
+value: false
+refs:
+  - zb: "0632.04005"
+    name: On completely Ramsey sets.
+---
+
+See the remark after Proposition 4 of {{zb:0632.04005}}.

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 set $[s, A]$, where $s$ and $A$ are finite and infinite, respectively, is clopen, and they form a basis by construction. Namely, if $x \notin [s, A]$, either $x \not \subseteq A \cup s$, in which case we let $t = x \cap (\min(x \setminus (A \cup s)) + 1)$, or $s \not \subseteq x$ or $\min(x \setminus s) \leq \max(s)$, in which case we let $t = (x \cap s) \cup \{\min(x \setminus s)\}$, or else $s \not \subseteq x$. Then $x \in [t, x]$ 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.
+---
+
+It is a theorem of Ellentuck that a countable union of nowhere dense sets is nowhere dense (Corollary 8 of {{zb:0292.02054}}), which is a strong form of being Baire.

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 open set has cardinality $\mathfrak{c}$ by construction.

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.