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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An Archimedean ordered field is an ordered field that satisfies the archimedean property.
The rational numbers are the initial ordered field, so for every ordered field there is a field homomorphism . is Archimedean if for all elements and , if , then there exists a rational number such that and .
Every Archimedean ordered field is a dense linear order. This means that the Dedekind completion of every Archimedean ordered field is the field of real numbers.
Every Archimedean ordered field is a differentiable space:
Let be an Archimedean ordered field. A function is continuous at a point if
is pointwise continuous in if it is continuous at all points :
The set of all pointwise continuous functions is defined as
Let be an Archimedean ordered field. A function is differentiable at a point if
is pointwise differentiable in if it is differentiable at all points :
The set of all pointwise differentiable functions is defined as
Archimedean ordered fields include
Non-Archimedean ordered fields include
The definition of the Archimedean property for an ordered field is given in section 4.3 of
Last revised on January 12, 2023 at 17:34:47. See the history of this page for a list of all contributions to it.