Let $B = \{[a, b] \mid \forall\, a, b \in \mathbb{R}, a < b\}$. Then $B$ a basis for some topology on $\mathbb{R}$.
Is it true that the set of all closed subsets in $\mathbb{R}$ generate a basis for some topology on $\mathbb{R}$? I know from previous assignments that the lower limit topology with half open intervals generates a topology but I doubt that the closed intervals would do the same. However I am not sure how to prove this.
Best Answer
As you have defined it, $B$ does not form a basis for a topology on $\mathbb{R}$. For $a<b<c$, we have $[a,b],[b,c]\in B$, but $\{b\}=[a,b]\cap[b,c]$, and no basis element both contains $b$ and belongs to $\{b\}$