On convergence to infinity

By a metric mode of convergence to infinity in a regular Hausdorff space X, we mean a sequence [F-k] Of closed subsets of X with Fk+1 subset of int F-k and boolean AND(k=1)(infinity) F-k = /, and a sequence (or net) [x(n)] in X is convergent to infinity with respect to [F-k] provided for each k, F-k contains x(n) eventually. Module a natural equivalence relation, these correspond to one-point extensions of the space with a countable base at the ideal point, and in the metrizable setting, they correspond to metric boundedness structures for the space. In this article, we study the interplay between these objects and certain continuous functions that may determine the metric mode of convergence to infinity, called forcing functions. Falling out of our results is a simple proof that each noncompact metrizable space admits uncountably many distinct metric uniformities.