Limit Superior and Limit Inferior - Functions From Metric Spaces To Metric Spaces

Functions From Metric Spaces To Metric Spaces

There is a notion of lim sup and lim inf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real sequence. Take metric spaces X and Y, a subspace E contained in X, and a function f : E → Y. The space Y should also be an ordered set, so that the notions of supremum and infimum make sense. Define, for any limit point a of E,

and

where B(a;ε) denotes the metric ball of radius ε about a.

Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have

and similarly

This finally motivates the definitions for general topological spaces. Take X, Y, E and a as before, but now let X and Y both be topological spaces. In this case, we replace metric balls with neighborhoods:

(there is a way to write the formula using a lim using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in is N ∪ {∞}.)