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
A field (as an algebraic structure) equipped with a metric is complete if the operations are continuous with respect to the induced topology and it is a complete metric space.
The most important case is the case of valued fields (i.e. multiplicatively normed fields).
Given a multiplicative discrete valuation on a field $k$, there is a topology on $k$ induced by the metrics induced by the valuation. A complete valued field is a field complete with respect to the valuation metric.
One of Ostrowski's theorems says that for $k$ a field complete with respect to an absolute value ${\vert - \vert}$ either the absolute value is archimedean, in which case $k$ is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.