Subsets of $\mathbb{R}$ 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!
 A: 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.
A: 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).
