For all topologies of R, are the open sets of R union of open intervals? My textbook gave the following proposition which followed the section on Euclidean topology:
"A subset S of R is open if and only if it is a union of open intervals"
but did not state whether this proposition is true only for the Euclidean topology or any topology on R. There are topologies on R that can be only closed intervals, but since infinite union of open interval can be closed intervals, I think the proposition is true for any topology on R right?
 A: In the topology consisting of the three open sets $\emptyset$, $\Bbb R$ and $\{0\}$, the open set $\{0\}$ is not a union of open intervals.
A: The book is defining the "standard" or "usual" topology on $\Bbb R. $
A topological space is a pair $(X,T)$ where $T$ is a collection of some or all of the subsets of $X,$ such that
(i). $X\in T$ and $\emptyset\in T,$ and
(ii). If $S$ is any sub-collection of $T$ then the common union $\bigcup S=\cup_{U\in S}\,U$ is in $T,$ and
(iii). If $S$ is any $finite$ subset of $T$ then the common intersection $\cap S$ is in $T.$
One example is $T=\{X, \emptyset\},$ called the coarse or anti-discrete topology. The opposite extreme is to have every subset of $X$ belong to $T,$ called the discrete or fine topology. Or, if you want, if $U$ is any  subset of $X,$ you can let $T=\{X,U, \emptyset\}.$
$T$ is called a topology on $X$ and the members of $T$ are called the open sets. "Open" is not an intrinsic property of any set. When you speak of an open set, your audience should already be aware of the topology that you are referring to.
