The basis for upper and lower limit topology is $\lbrace (a,b]\mid a<b\rbrace$ and $\lbrace [a,b)\mid a<b\rbrace$ respectively. But the basis of discrete topology on $\mathbb{R}$ is $\lbrace\lbrace x\rbrace\mid x\in\mathbb{R}\rbrace$. How can I show that they are the same? I can not find a similar question to this and I have no ideas how to show this. Any hints would be most appreciated.
Union of Upper Limit Topology and Lower Limit Topology is Discrete Topology
general-topologysorgenfrey-line
Best Answer
If all intervals of the form $(a,b]$ or $[a,b)$ are open, then for any $x\in \Bbb R$, we have that $$\{x\}=(x-1,\,x]\,\cap\, [x,\,x+1)$$ is open.