Back-and-forth argument

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Problem III.4.7 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.4 "Continuous Mappings"

Let $X, Y \subset (0,1)$ be countable sets that are everywhere dense in $\left[0,1\right]$ . Construct a diffeomorphism of the interval $\left[0,1\right]$ (a mapping onto itself, continuously differentiable together with its inverse mapping) carrying $X$ onto $Y$ .

Problem I.5.13 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter I.5 "Ordinal Numbers"

(a)(b)

Prove that any countable linearly ordered set $A$ has order type $\eta$ if and only if $A$ satisfies the following conditions:

(a)

$A$ has no least and no greatest element;

(b)

for any $x, y \in A$ such that $x < y$ , there exists $z \in A$ such that $x < z < y$ (such an order is called dense).

Problem I.5.14 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter I.5 "Ordinal Numbers"

Prove that every countable linearly ordered set is similar to some subset of the set $\mathbb {Q}$ of rational numbers.

References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.

[LaMa02] Lavrov I.A., Maksimova L.L. Задачи по теории множеств, математической логике и теории алгоритмов. 5-е изд., Физматлит, 2002.