I was wondering if someone could help with the following problems:
(a) Give an example of a topological space $(X,\mathcal{T})$ and a subset $A$ of $X$ which is both open and closed.
(b) Give another example where $A$ is neither empty nor the whole of $X$.
(c) Give an example of a topological space $(X,\mathcal{T})$ and a subset $A$ of $X$ which is neither open nor closed.
My answers:
(a) Let $X$ be any set with any topology $\mathcal{T}$, let $A=\emptyset$.
(b) Let $X=(1,2)\cup(3,4)$, $A=(1,2)$ and $\mathcal{T}$ be the relative topology inherited from the usual topology on $\mathbb{R}$.
(c) Let $X=\mathbb{R}$ with the usual metric and let $U_{n}=(0,1+\frac{1}{n})$.Then take $A=\displaystyle\bigcap_{n=1}^{\infty} U_{n}=(0,1]$.
Best Answer
Your answers are totally correct. I fail to see, however, why you decided to write your answer for (c) as an infinite intersection.
To prove that $A = (0,1]$ is not open, it suffices to show that $1$ has no open neighborhood in $A$. To prove that $A$ is closed, it suffices to show that $0$ has no open neighborhood outside of $A$.