In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
, a continuous function is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
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. These continuous functions are often used in
cofinalities and
cardinal numbers.
A
normal function is a function that is both continuous and
strictly increasing.
References
*
Thomas Jech. ''Set Theory'', 3rd millennium ed., 2002, Springer Monographs in Mathematics, Springer,
Set theory
Ordinal numbers
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