Just started

[Abaddon_ire]

P.S. Since my goal is to learn material covered in an intro to real analysis course, would you recommend that I also study metric spaces or is this topic more advanced than the material that is usually covered in an intro level course? Thanks again for your time and attention.

In my opinion, metric spaces are better studied as a precursor to topology. But this depends on how it is done and what the later goals are. Other mathematicians disagree and have good reasons for doing so.

It isn't so much that the material is more advanced as it is more abstract. Many of the proofs from real analysis carry over *almost* verbatim to metric spaces. And while others fail, that failure can clarify what is going on in the real analysis. The nice thing is that doing metric spaces opens up the analysis of function spaces and puts uniform convergence of functions in its 'proper' setting.

On the other hand, questions of differentiability and integrability simply don't arise in metric spaces. The focus is  on issues related to continuity between very general spaces. So if your goal is to understand derivatives and integrals, it is probably best to wait for metric spaces until after real analysis. If, instead, you want to understand questions concerning continuity, uniform continuity, Cauchy sequences, compactness, and completeness into proper perspective, do some metric spaces. Metric spaces also open up the study of Banach spaces and Hilbert spaces, which are large subjects in themselves.

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So far, to me you seem genuine. But that is provisional. In other posts, it appears to me you are an archetype of seeking a PhD paper.

There is nothing per se wrong with that. But if that is your goal, honesty would yield a more productive paper.

If I stumble across a good *physics* PhD topic, and I could get one of the local physics profs to be my advisor, I wouldn't complain. But I don't expect that to happen.

Do you really have a problem with PhD candidates wanting to write about this forum?

Not in the slightest.