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Give examples of subsets of $\mathbb{R}$ that are:

a) Infinite, but not connected

My attempt: $(-\infty,1)\cup (2,\infty)$, because it can be represented as the union of two open subsets, and it is infinite

b) Bounded and countable

My attempt: $\mathbb{Z}\cap[0,10]$ because it is bounded by $0$ and $10$, and contains 10 elements, so it must be countable

c) Bounded and uncountable

My attempt: [0,10] because an interval of $\mathbb{R}$ is uncountable (right?)

d) Closed but not compact

My attempt: All of $\mathbb{R}$? We're looking for something closed but not bounded to find something not compact, correct?

e) Dense but not complete

My attempt: $\mathbb{Q}$ because the closure of $\mathbb{Q}$ is equal to $\mathbb{R}$, but it does not contain all its limits, e.g. $\sqrt{2}$.

Any input appreciated!

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  • $\begingroup$ a) Why are you using $\infty$? Isn't the set $(1,2)$ already infinite, even uncountably infinite? $\endgroup$ – Dietrich Burde Jan 31 '17 at 15:01
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    $\begingroup$ b) Some authors use "countable" to mean countably infinite, so check your source. In any case "bounded and countably infinite" subsets of $\mathbb{R}$ do exist and are a bit more interesting than merely finite subsets. $\endgroup$ – hardmath Jan 31 '17 at 15:09
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    $\begingroup$ @Hardmath I wanted to say, without $\infty$ in both parts, so $(1,2)\cup (3,4)$. $\endgroup$ – Dietrich Burde Jan 31 '17 at 15:17
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Everything looks fine, except I think that in problem $b)$ they want a set that can be bijected with $\mathbb N$.

I suggest $\mathbb Q \cap [0,1]$


Under further inspection I am sure, countable sets need to have the exact same cardinality as $\mathbb N$, the term used for a countable or finite set is denumerable or at most countable.

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    $\begingroup$ Some people define countable to include finite, some require that it is infinite. This works for countably infinite. $\endgroup$ – Ross Millikan Jan 31 '17 at 15:17
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Fine, up to details that might add value to your work.

In (a) you should mention that the set is the union of two open disjoint sets. This is crucial: also $(0,2)\cup(1,3)$ is the union of two open sets, but it's connected.

In (b) you probably should show an infinite countable and bounded set (but this depends on what you mean by countable): the numbers of the form $1/n$ (for $n$ a positive integer) make for an easy example; if you also add $0$, you get an example of a compact countable set.

(c) and (d) are fine. For (c) you might consider $(-\pi/2,\pi/2)$, instead, so it's easier to exhibit a bijection with $\mathbb{R}$, if requested: the tangent function.

Also (e) is fine, but you just have to mention $\mathbb{Q}$ is dense. Since it is not the whole of $\mathbb{R}$ it cannot be complete (a complete subset must be closed).

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