In the lower limit topology, I am trying to get the interior and closure of the set $(0,1]$. I think its interior is $(0,1)$.
I am also confused about the closure since $[0,1]$ wouldn't be closed, and $[0,1)$ which is open does not contain $1$. What should I do to get the closure?
Thanks
Best Answer
The interior is indeed $(0,1)$. This is open because $(0,1)=\bigcup_{n \in \Bbb{N}}[\frac{1}{n},1)$.
The closure is $[0,1]$. It is closed because for every $x \notin [0,1]$, $[x,0)$ or $[x,\infty)$ are opens not intersecting $[0,1]$ (depending on whether $x<0$ or $x>1$). Since it has only one point more than $(0,1]$ and that $(0,1]$ isn't closed, it is the closure.