S60: homogeneity, cut points and more

spaces/S000060/properties/P000056.md

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+---
+space: S000060
+property: P000056
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+The set $\mathbb R\setminus \mathbb Q$ is closed with empty interior, so are the sets $\{x\}$ for $x\in\mathbb Q$. The whole space is clearly the union of this countable family.

spaces/S000060/properties/P000082.md

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+---
+space: S000060
+property: P000082
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+For any $x\in\mathbb R$ the topology on $\mathbb Q\cup\{x\}$ coincides with the one induced by the standard Euclidean metric.

spaces/S000060/properties/P000086.md

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+---
+space: S000060
+property: P000086
+value: true
+refs:
+- mathse: 5013911
+  name: "Pointed rational extension of reals is a homogeneous space"
+---
+
+See {{mathse:5013911}}.

spaces/S000060/properties/P000089.md

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+---
+space: S000060
+property: P000089
+value: false
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+It is evident that $x\mapsto x+1$ is a homeomorphism and has no fixed point.

spaces/S000060/properties/P000132.md

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+---
+space: S000060
+property: P000132
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+Let $A$ be a closed set. Clearly, $A_0:=A\cup \mathbb Q$ is open (this is true for arbitrary set $A$). Let $\{q_n\}_{n=1}^\infty$ be an enumeration of
+points from $A_0\setminus A$. Then for every $n\geq 1$ the set $A_n:=A_0\setminus\{q_k:k=1,\ldots,n\}$ is open (by {P2})
+and $A=\bigcap_{n=1}^\infty A_n$.

spaces/S000060/properties/P000189.md

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+---
+space: S000060
+property: P000189
+value: false
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+The space can be partitioned into a countable family of nonempty closed sets, namely the set $\mathbb R\setminus \mathbb Q$ and singletons of rational numbers.

spaces/S000060/properties/P000205.md

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+---
+space: S000060
+property: P000205
+value: true
+refs:
+- doi: 10.1007/978-1-4612-6290-9_6
+  name: Counterexamples in Topology
+---
+
+$X$ is {P36}: see item #4 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.
+
+For every $x\in\mathbb R$ the subsets $(-\infty,x)$ and $(x,+\infty)$ are open hence $x$ is a cut point.