definition and example of standard topology

Question

Let $X$ be a topological space. Then

trivial topology $T$ is $\{\phi,X\}$

discrete topology $T$ is the family of all subsets of $X$.

standard topology $T$ is the collection of all open intervals of $X$?

I understand the trivial and discrete topologies but I don't know how to approach to the standard topology. Can someone give me a simple example of standard topology?

Answer 1

There is no such thing as the standard topology on any set $X$. If $X=\mathbb R$, then the standard topology is the topology whose open sets are the unions of open intervals. More generaly, if $X\subset\mathbb R$, then the standard topology is the topology whose open sets are the unions of sets of the type $(a,b)\cap X$, with $a,b\in\mathbb R$ and $a<b$.