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
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 ord ...
then ''s'' is continuous if ''s'': ''γ'' → range(''s'') is a
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
when the sets are each equipped with the
order topology
In mathematics, an order topology is a specific 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, ...
. These continuous functions are often used in
cofinalities and
cardinal numbers
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case ...
.
A
normal function
In axiomatic set theory, a function is called normal (or a normal function) if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:
# For every limit o ...
is a function that is both continuous and
strictly increasing
In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusiv ...
.
References
*
Thomas Jech
Thomas J. Jech (, ; born 29 January 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 2000 is at thInst ...
. ''Set Theory'', 3rd millennium ed., 2002, Springer Monographs in Mathematics, Springer,
Set theory
Ordinal numbers
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