# definition and example of standard topology

Let $X$ be a topological space. Then

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

discrete topology $T$ = family of all subsets of $X$

standard topology $T$ = ??collection of all open intervals of X??

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

• The trivial topology is also called the indiscrete topology. FYI. – Henno Brandsma Apr 28 '18 at 23:10
• The trivial topology is also called the anti-discrete, or coarse topology. – DanielWainfleet Apr 29 '18 at 4:30

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 $(a,b)\cap A$, with $a,b\in\mathbb R$ and $a<b$.