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

spaces/S000216/README.md

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+---
+uid: S000216
+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)
+---
+
+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 $\mathcal{E} = \{E_r : r\in\mathbb{R}\setminus\mathbb{Q}\}$. Let $E'$ be the set of limit points for a subset $E$ of {S108}. Then $E'\neq \emptyset$ for $E \in\mathcal{E}$. For each $E\in\mathcal{E}$ pick some $p_E\in E'$.
+
+Katětov's non-normal subspace of $\beta\mathbb{N}$ is the space $X=\mathbb{N}\cup D$ where $D = \{p_E : E\in\mathcal{E}\}$.
+
+Constructed in exercise 6Q of {{doi:10.1007/978-1-4615-7819-2}}.

spaces/S000216/properties/P000006.md

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+---
+space: S000216
+property: P000006
+value: true
+---
+
+$X$ is contained in {S108} and {S108|P6}.

spaces/S000216/properties/P000007.md

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+---
+space: S000216
+property: P000007
+value: false
+---
+
+$D$ is a closed discrete subspace of $X$ of size $\mathfrak{c}$, so there are $2^\mathfrak{c}$ continuous real-valued functions on $D$ and at most $\mathfrak{c}$ continuous real-valued functions on $X$ since {S216|P26}.
+
+If $X$ were $T_4$ then from Tietze extension theorem we would obtain $2^\mathfrak{c} \leq \mathfrak{c}$, contradiction. So $X$ is not $T_4$.

spaces/S000216/properties/P000026.md

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+---
+space: S000216
+property: P000026
+value: true
+---
+
+$\mathbb{N}\subseteq X$ is countable and dense.

spaces/S000216/properties/P000049.md

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+---
+space: S000216
+property: P000049
+value: true
+---
+
+$X$ contains {S2} and is therefore a dense subspace of {S108} and {S108|P49}.

spaces/S000216/properties/P000065.md

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+---
+space: S000216
+property: P000065
+value: true
+---
+
+$X$ is in bijection with $\mathbb{R}$.

spaces/S000216/properties/P000112.md

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+---
+space: S000216
+property: P000112
+value: true
+---
+
+Extend $\varphi$ to $X$ so that if $E = E_r = \{\varphi^{-1}(s_n) : n\in\omega\}$ and $s_n\to r$ then $\varphi(p_E) = r$. If $\varphi(p_E)\in U$ where $U\subseteq \mathbb{R}$ is open, find $N$ such that $s_n\in U$ for $n\geq N$. Since {S216|P49}, $V =\overline{E}\setminus\varphi^{-1}(\{s_1, s_2, ..., s_N\})$ is an open neighbourhood of $p_E$ and $\varphi(V)\subseteq U$. So $\varphi:X\to\mathbb{R}$ is a continuous injection, hence $X$ is submetrizable.