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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Once infinitesimal analysis was a synonym for “infinitesimal calculus”, hence for the foundations of the theory of differentiation and integration, but it has come to refer to the formulation of analysis using explicit infinitesimal objects, in contrast to epsilontic analysis.
Formalisms that make infinitesimal analysis precise include
synthetic differential geometry with nilpotent infinitesimals
nonstandard analysis with non-nilpotent infinitesimals