What is the definition of open-interval in topology? I just found that open-set is open-interval.
In lower limit topology, [a.b) is one open set. is it open interval in lower limit topology?
If so, general definition of open interval in real analysis is not applied in topology?
 A: The foundation for all of topology is open sets. Topologists take a set, and then decide which of its subsets they want to be "open." The collection of open sets is called a topology. Usually, when our set is $\mathbb R$, the sets that are open are unions of open intervals, $(a,b)$, meaning "all of the numbers greater than $a$ and less than $b$." When all of our open sets can be written as unions of open intervals, we are working with the standard topology on $\mathbb R$. We can choose to work with other, topologies, though. The lower limit topology is a non-standard topology in $\mathbb R$ which is still interesting to study. So to answer your question, topologists do not limit themselves to studying just one topology. Usually, we do study the sets that are considered open in real analysis, but often, there are other interesting ways (like in the lower limit topology) to define an open set.
In general, there is no concept of "interval" in topology, only in specialized cases do we have intervals. These specialized cases are when the topology is derived from an order on the set. Since the real numbers are ordered, we can talk about intervals. If your topology is the lower-limit topology, $[a,b)$ is an open set, and it is an interval, but it should not be called an open interval. This term is reserved for sets of the form $(a,b)$, regardless of their topology.
A: An open set might not be an open interval. Consider $\bigcup_{x \in \mathbb{Z}} ]x-1/2,x+1/2[$.
The definition of open interval is connected with ordering. If $(X,\leq)$ is a totally ordered set, an open interval is ($a,b \in X, a \leq b$)
$$ \{x \in X\mid a < x < b\}
$$
The topology generated by these sets are called the order topology. It is the usual topology on $\mathbb{R}$.
