Formal ball

In topology, a branch of mathematics, a formal ball is an extension of the notion of ball to allow unbounded and negative radius. The concept of formal ball was introduced by Weihrauch and Schreiber in 1981 and the negative radius case by Tsuiki and Hattori in 2008.
Specifically, if is a metric space then an element of is a formal ball, where is the set of nonnegative real numbers. Elements of are known as generalized formal balls.
Formal balls possess a partial order defined by if.
Generalized formal balls are interesting because this partial order works just as well for as for, even though a generalized formal ball with negative radius does not correspond to a subset of.
Formal balls possess the Lawson topology and the Martin topology.