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. enter i

I would like to prove that botn definitions are equivalent.

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    $\begingroup$ It doesn't need to be countable. $\endgroup$ – user230734 Aug 26 '15 at 20:49
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    $\begingroup$ You want a proof of a definition? $\endgroup$ – user642796 Aug 26 '15 at 20:55
  • $\begingroup$ 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. $\endgroup$ – Bruno Schiavo Aug 26 '15 at 21:01
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    $\begingroup$ 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. $\endgroup$ – Silvia Ghinassi Aug 26 '15 at 21:03
  • $\begingroup$ Sorry for the ambiguity. I've just edited it to make it more precise. $\endgroup$ – 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.

  • $\begingroup$ Why can this Union be infinite, and why must the Intersection of sets of interior points be finite? $\endgroup$ – Bruno Schiavo Aug 26 '15 at 21:20

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