(Topology) In the real line $R^1$ a set is open if and only if it is a union of open segments (a,b).

I am reading Rudin's "Real and Complex Analysis" and came across the statement,
"In the real line $R^1$ a set is open if and only if it is a union of open segments (a,b)."

I have two questions:
1. A subset $A$ of $X$ may be open with respect to a topology $\tau_1$ in $X$, but not open with respect to another topology $\tau_2$ in $X$. Therefore, what does it mean to say that, a set is open in the real line $R^1$, without specifying underlying topology?
2. I see that a collection of all sets which are arbitrary unions of open interval consists a topology $\tau$ in $R^1$. So arbitrary union of open interval is open in X with topology $\tau$.(if part, which is straight forward.) But how can I show that arbitrary open set in X with respect to some random topology $\tau'$ can be expressed as a union of open intervals?(only if part) I think this question is related to the first question since I need clarification regarding the meaning of an open set in X means without any topology specified.

I do realize that $R^1$ is a metric space, and the statement is perfectly clear with the definition of open set in a metric space. But is there a way to understand the above statement without relying on the concept of metric space?

• 1. There is a "standard" topology on $\Bbb R$. – Lord Shark the Unknown Mar 17 '18 at 3:28
• MathJax tip: \Bbb{R} = $\Bbb{R}$ – gen-z ready to perish Mar 17 '18 at 3:31
• 2. The quoted statement is false if we consider a topology different from the standard one. Indeed, the statement specifies the basis with which the corresponding topology on $\mathbb{R}$ is generated. – Sangchul Lee Mar 17 '18 at 3:31
• Intentionally obtuse. -1. – MPW Mar 17 '18 at 3:39
• There are two natural ways to define a topology, namely via a metric from the absolute value, or via its linear order (the order topology). They both coincide for the reals. Rudin's statement holds in general for linearly ordered spaces without min or max, so should IMHO not be seen as a "metric" rather than as an "order" statement. – Henno Brandsma Mar 17 '18 at 16:28

Rudin is saying that in the standard topology on $\Bbb R$ induced by the metric $d(x,y) = |x-y|$, every open set can be written as a union of intervals of the form $(a,b)$.
Whenever they do not specify the topology on $\mathbb R$, they mean the Standard Topology on $\mathbb R$