I am currently preparing for an exam in functional analysis and I am working through some exercises. However, I do not have the answers for them. Usually I do not need them, but I am still very unconfident with topological terms and the properties of sets, so I hope you can simply tell me, whether my ansers here are correct:
Question is the following: Determine whether the properties (open, closed, compact, bounded) are true for the following sets using the standard Euclidean topology over $\mathbb{R}$:
$ \{\frac 1 n + 1 | n \in \mathbb{N}\} $ My thoughts here: All elements are obviously between 2 and 1 (1 excluded), so the set is bounded. However, my problems are to decided whether it is open or closed: For every element of the set, there is no $\epsilon$-environment so that all elements in this environment are also part of the set. For the complement you can always find such an environment. Thus this set is closed. Since it is closed and bounded, it is also compact. True?
Next doubtful point: $\mathbb{Q}$ This is obviously not bounded. Again, you are not able to find $\epsilon$-environments around points in $\mathbb{Q}$. However, you are also not able to find $\epsilon$-environments around points in the complement $\mathbb{R}$ \ $\mathbb{Q}$. Thus it is both open and closed.
Thanks for your comments :)