# how to prove that the topology generated by the left-closed intervals is finer than the usual topology

The idea is to prove that the open intervals (like $$]a,b[$$) are contained in the topology of the left-closed sets ($$[a,b[$$), but I cannot see a way of generating open sets from half-closed sets. (Same goes for the right-closed sets)

$$]a,b[ = \bigcup_{n\in\mathbb{N}} [a + \frac{1}{n},b[$$
Hint: Note that the sequence $$\frac1n\to 0$$, but $$0\notin\{\frac1n:n\in\mathbb N\}$$.