Katetov's non-normal subspace of $\beta\mathbb{N}$

spaces/S000208/README.md

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+---
+uid: S000208
+name: Katětov's non-normal subspace of $\beta\mathbb{N}$
+refs:
+  - doi: 10.1007/978-1-4615-7819-2
+    name: Rings of Continuous Functions (Gillman & Jerison)
+---
+
+Constructed in exercise 6Q of {{doi:10.1007/978-1-4615-7819-2}}. Fix a bijection $\varphi:\mathbb{N}\to\mathbb{Q}$. For each irrational $r$ fix a sequence of rational numbers $s_n\to r$, and let $E_r = \{\varphi^{-1}(s_n) : n\in\mathbb{N}\}$. Let $\mathcal{E} = \{E_r : r\in\mathbb{R}\setminus\mathbb{Q}\}$. Then $\mathcal{E}$ is an almost disjoint family on $\mathbb{N}$ of size $\mathfrak{c}$.
+
+Let $E'$ be the set of limit points of $E$ in {S108}. Then $E'\neq \emptyset$ and $E_1'\cap E_2' = \emptyset$ for $E, E_1, E_2\in\mathcal{E}$. For each $E\in\mathcal{E}$ take $p_E\in E'$ and let $\Pi = \mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$.
+
+Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $\Pi$.
+
+

spaces/S000208/properties/P000006.md

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+---
+space: S000208
+property: P000006
+value: true
+---
+
+Is a subspace of {S108} and {S108|P6}.

spaces/S000208/properties/P000007.md

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+---
+space: S000208
+property: P000007
+value: false
+---
+
+$D$ is a closed discrete subspace of $\Pi$ of size $\mathfrak{c}$. Since there is $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $\Pi$, from Tietze extension theorem $\Pi$ cannot be $T_4$.

spaces/S000208/properties/P000049.md

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+---
+space: S000208
+property: P000049
+value: true
+refs:
+- zb: "0684.54001"
+  name: General Topology (Engelking, 1989)
+---
+
+It's a dense subspace of {S108} and {S108|P49}. A dense subspace of extremally disconnected space is extremally disconnected (see {{zb:"0684.54001"}} exercise 6.2.G.c).

spaces/S000208/properties/P000162.md

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+---
+space: S000208
+property: P000162
+value: true
+refs:
+  - doi: 10.1007/978-1-4615-7819-2
+    name: Rings of Continuous Functions (Gillman & Jerison)
+---
+
+Extend $\varphi$ to $\Pi$ so that if $E = E_r$ and $s_n\to r$ then $\varphi(p_E) = \lim_{n\to\infty} \varphi(s_n)$. If $\varphi(p_E)\in U$ where $U\subseteq \mathbb{R}$ is open, find $N$ such that $s_n\in U$ for $n\geq N$, then $V =\overline{E}\setminus\varphi^{-1}(\{s_1, s_2, ..., s_N\})$ is an open neighbourhood of $p_E$ and $\varphi(V)\subseteq U$. This shows that $\varphi:\Pi\to\mathbb{R}$ is continuous. By definition it's clearly a bijection. From corollary 8.18 of {{10.1007/978-1-4615-7819-2}}, $\Pi$ is realcompact.