In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a continuous function is a sequence of
ordinals such that the values assumed at limit stages are the limits (
limit suprema and limit infima) of all values at previous stages. More formally, let γ be an ordinal, and
be a γ-sequence of ordinals. Then ''s'' is continuous if at every limit ordinal β < γ,
:
and
:
Alternatively, if ''s'' is an increasing function then ''s'' is continuous if ''s'': γ → range(s) is a
continuous function when the sets are each equipped with the
order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, th ...
. These continuous functions are often used in
cofinalities and
cardinal numbers
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
.
A
normal function is a function that is both continuous and
increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
.
References
*
Thomas Jech
Thomas J. Jech ( cs, Tomáš Jech, ; born January 29, 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years.
Life
He was educated at Charles University (his advisor was Petr Vopěnka) and from ...
. ''Set Theory'', 3rd millennium ed., 2002, Springer Monographs in Mathematics,Springer,
Set theory
Ordinal numbers
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