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Forgive me as this is a question inspired by my first computer science algorithms class. I am a math major so when I learned of the big-theta operation on the space of continuous functions I began to wonder what the space of equivalence classes looks like given the big-theta equivalence relation. Is there a countable amount of equivalence classes? I have found any good answers for this question online.

Jonah Weinbaum
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    Consider $x^\alpha$ where $\alpha\in\mathbb{R}^+$ – saulspatz Jan 12 '21 at 22:00
  • Yeah thats makes sense to show its not countable. I guess I'm wondering if this space is a little "nicer" than the space of all continuous functions in any notable ways. – Jonah Weinbaum Jan 12 '21 at 22:02
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    If you keep the CS/algorithms motivation, that seems relevant (time hierarchy theorem) and could be an interesting addition: https://en.wikipedia.org/wiki/Time_hierarchy_theorem – Clement C. Jan 12 '21 at 22:09

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