A collection of subsets of $\mathbb{R}$ that is not a topology I want to show that the collection of sets of the form: $$\{(-\infty, x] : x \in \mathbb{R}\}$$ together with the empty set and $\mathbb{R}$, is not a topology for $\mathbb{R}$. But if I take the infinite union of sets of the form $\{(-\infty, x] : x \in \mathbb{R}\}$, this is not itself a set of the form $\{(-\infty, y] : y \in \mathbb{R}\}$, where $y$ is the maximum of all $x$? And if I take the finite intersection of this sets, this is not again, a set of the form $\{(-\infty, x] : x \in \mathbb{R}\}$, where $x= \text{min}(x_1,...,x_n)$? 

Take the union $$\bigcup_{x<y} (-\infty,x],$$ this is $(-\infty,y)$ which is not in your collection of open sets. 
