# Axiomatic proof that all points of an open set are interior points

In "Principles of Mathematical Analysis, Rudin the following definition (f) to open sets: a set is open if all of its points are interior points Sidney Morris' Topology Without Tears, however, defines open sets as members of a topology, a set with the following axioms: Si. I would like to prove that botn definitions are equivalent.

• It doesn't need to be countable. – user230734 Aug 26 '15 at 20:49
• You want a proof of a definition? – user642796 Aug 26 '15 at 20:55
• I've described two definitions of open sets: (1) open sets as sets of interior points and (2) open sets as members of a topology. I want to show that both are equivalent. – Bruno Schiavo Aug 26 '15 at 21:01
• That is not what you have written in the question. You are being downvoted because it's unclear what you are asking, maybe you want to edit. – Silvia Ghinassi Aug 26 '15 at 21:03
• Sorry for the ambiguity. I've just edited it to make it more precise. – Bruno Schiavo Aug 26 '15 at 21:13

Let $E$ a set such that every point of $E$ is an interior point of $E$.
By definition of an interior point $\forall x\in E$, there exists an open set $O_x$ such that $x\in O_x$ and $O_x\subset E$.
Then: $$E=\bigcup_{x\in E}O_x$$
$E$ is open as an union of open sets.