Continuous function (set theory)

In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the union or supremum of all ordinals in previous ones. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and More formally, let γ be an ordinal, and $s := \langle s_{\alpha}| \alpha < \gamma\rangle$ be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ $, s_{\beta} = \sup\{s_{\alpha}| \alpha < \beta\}.$ Alternatively, s is continuous if s: γ → range(s) is a continuous function when the sets are each equipped with the order topology. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.

References

Thomas Jech. Set Theory, 3rd millennium ed. , 2002, Springer Monographs in Mathematics,Springer, ISBN 3-540-44085-2

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