Thread: Countinuum Hypothesis in Analysis.

Is there any important result from Countinuum hypothesis in analyis? For example, Zorn's Lemma is equivalent to saying that every sequence with an upper bound has a least upper bound- which is the foundation of all analyis. But what about Countinuum hypothesis?

Originally Posted by ThePerfectHacker

Is there any important result from Countinuum hypothesis in analyis? For example, Zorn's Lemma is equivalent to saying that every sequence with an upper bound has a least upper bound- which is the foundation of all analyis. But what about Countinuum hypothesis?

I'd say the answer is no at least if we are talking about up to first-year graduate-level analysis. The textbooks on analysis and topology that I have mention the continuum hypothesis perhaps once and never about anything important. But Zorn's Lemma or the equivalent Axiom of Choice are indeed fundamental and used many times.

Nice topic! And I just found out

Two applications for these theorems:

Zorn's Lemma: The Hahn-Banach extension theorem.

Axiom Of Choice: Tychonoff's Theorem.

I am hoping that there is some simple theorem in analysis equivalent to CH which would lead to RH.