Why is $(a,b)$ open on the lower limit topology of $\mathbb{R}$?

Question

In the book of General Topology by Munkres, on page 104, it is given that

However, as far as I know, the lower limit topology $\tau_l$ corresponds to the intervals of the form $[a,b)$ where $a < b$. So how can $(a,b)$ be open in this topology?

Answer 1

$(a,b)=\bigcup_{n=1}^\infty[a+1/n,b)$

Answer 2

Not every open set in the lower limit topology has the form $[a,b)$. Such sets just form a basis for the topology, so every open set is a union of sets of this form. You can write an interval $(a,b)$ as the union of the sets $[c,b)$ for all $c>a$, so $(a,b)$ is open in the lower limit topology.

Answer 3

$[x,b)\subset (a,b)$ for every $x\in (a,b)$ and $[x,b)$ is open.