Topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable? Could someone list the topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable that a new math graduate student should be familiar with? Also could someone list the recommended/standard study materials for these topics? Thanks in advance!
Edit1: Precisely I wish to ask "the topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable that new math graduate students are expected to be already familiar with". Thanks for the comment for clarification by Mike Haskel.
Edit2: Sorry. This question is a cross-posting. The same question is asked in Reddit r/math here: https://redd.it/3ppxn9
 A: I'm going to assume you're asking what a new student might be expected to know when they arrive at graduate school. A good start would be familiarity with the following topics, to the point where the student can do proof-based exercises without assistance.
Walter Rudin gives a good treatment of real analysis at this level in his Principles of Mathematical Analysis (a.k.a. "Baby Rudin," ISBN 978-0070542358).
Metric Spaces


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*Convergence, Cauchy sequences, and completeness.

*Continuity vs. uniform continuity vs. Lipschitz continuity.

*Pointwise vs. uniform convergence.


Differentiation


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*Derivatives of functions $\mathbb{R^n} \to \mathbb{R^m}$ as linear maps.

*Multivariable Taylor approximation.

*Inverse and implicit function theorems.


Integration


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*Formal development of Riemann integration.


Topology of $\mathbb{R}$


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*$\mathbb{R}$ is the unique realization of a certain list of axioms.

*Bolzano-Weierstrass theorem (closed and bounded iff sequentially compact).

*Heine-Borel theorem (closed and bounded iff topologically compact).

*Topological connectedness.

