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

  • 1
    $\begingroup$ $(0,1)$ is the union of all the basic opens $[1/n,1)$ for $n\geq 2$, and therefore perfectly open. $\endgroup$
    – Arthur
    Commented Apr 30, 2016 at 22:00

1 Answer 1


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.


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