Traits for S140: \mathbb{R} extended by a point with cocountable open neighborhoods (#1656)

Author: mathmaster13

spaces/S000140/README.md

@@ -1,13 +1,13 @@
 ---
 uid: S000140
-name: Real numbers extended by a point with co-countable open neighborhoods
+name: $\mathbb{R}$ extended by a point with co-countable open neighborhoods
 refs:
 - mathse: 4850979
   name: Answer to 'Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces'
 - mathse: 4854178
   name: What are the compactness properties of $\mathbb R$, extended by a point with co-countable open neighborhoods?
 ---
 
-Let $X=\mathbb R\cup \{\infty\}$, with $\mathbb R$ having the Euclidean topology and open in $X$, and open neighborhoods of $\infty$ given by sets of the form $U\cup\{\infty\}$, where $U\subseteq \mathbb R$ is co-countable and open in the Euclidean topology. 
+Let $X=\mathbb R\cup \{\infty\}$. Let $\mathbb R$ be open in $X$ and have the topology of {S25}, and let open neighborhoods of $\infty$ be given by sets of the form $U\cup\{\infty\}$, where $U\subseteq \mathbb R$ is co-countable and open in the Euclidean topology. 
 
 Constructed in {{mathse:4850979}} as an example of a space that is {P173} and {P81}, yet fails to be {P79}, yielding a counterexample to a natural analogue of {T211}.  Elaborated on in {{mathse:4854178}}.

spaces/S000140/properties/P000073.md

@@ -0,0 +1,11 @@
+---
+space: S000140
+property: P000073
+value: true
+---
+
+Let $S \subseteq X$ be a nonempty irreducible ({P39}) set.
+The subspace $\mathbb R$ is open in $X$ and Hausdorff.
+It follows that $S$ cannot contain more than one point from $\mathbb R$, and thus $S$ is finite.
+On the other hand, $X$ is {P2}.
+Consequently $S$ is discrete, which implies it must be a singleton.

spaces/S000140/properties/P000189.md

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+---
+space: S000140
+property: P000189
+value: true
+---
+
+$\mathbb R$ is a dense subset of $X$ and {S25|P189}.

spaces/S000140/properties/P000204.md

@@ -0,0 +1,11 @@
+---
+space: S000140
+property: P000204
+value: false
+---
+
+The point $\infty$ is not a cut point, since $X \setminus \{\infty\}=\mathbb R$ is connected.
+And if $x\in\mathbb R$, the intervals $(\leftarrow, x)$ and
+$(x, \to)$ in $\mathbb{R}$ are connected,
+hence so are their closures $(\leftarrow, x)\cup\{\infty\}$ and $(x, \to)\cup\{\infty\}$.
+Since the two closures have a point in common, their union $X\setminus\{x\}$ is also connected.

spaces/S000140/properties/P000210.md

@@ -0,0 +1,10 @@
+---
+space: S000140
+property: P000210
+value: true
+refs:
+- mathse: 5126685
+  name: Do any of the Arkhangel'skii $\alpha_i$ properties hold for $\mathbb{R}$ extended by a point with co-countable open neighborhoods?
+---
+
+See {{mathse:5126685}}.

spaces/S000140/properties/P000219.md

@@ -0,0 +1,9 @@
+---
+space: S000140
+property: P000219
+value: false
+---
+
+$X$ and its subspace $\mathbb R$ have the same cardinality.
+But they are not homeomorphic:
+{S25|P3} and {S140|P3}.