Show $(0,1)$ is open but not closed in the Lower Limit Topology.
I know that $[a,b)$ is open and closed in the lower limit topology, but I am not sure how to prove this one.
Thanks for any help.
[Math] Lower Limit Topology
general-topology
general-topology
Show $(0,1)$ is open but not closed in the Lower Limit Topology.
I know that $[a,b)$ is open and closed in the lower limit topology, but I am not sure how to prove this one.
Thanks for any help.
Best Answer
To show $(0,1)$ is open in LL topology note the following $$ x \in [x,1) \subset (0,1) ~~\text{for all $x \in (0,1)$}. $$
Hence $(0,1)$ is open in LL topology.
To show $(0,1)$ not closed in LL topology, we shall show that closure of $(0,1)$ in LL topology is not $(0,1)$.
Take any neighborhood $N$ of $0$. There exists $a>0$ such that $0 \in [0,a) \subset N$
Hence $N \cap (0,1)$ is not empty.
Hence $0 \in \overline{(0,1)}$. But $0 \notin (0,1)$.
Hence $(0,1)$ is not closed in LL topology.
Hope this helps.
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