analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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symmetric monoidal (∞,1)-category of spectra
Any non-trivial absolute value on the rational numbers is equivalent either to the standard real absolute value, or to the p-adic absolute value.
(Ostrowski)
Any complete archimedean valued field is, as an associative algebra and as a topological space, isomorphic to either the real numbers or the complex numbers.
In the course notes
Ostrowki’s theorem is discussed around p. 23.