topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, a formal ball is an extension of the notion of

ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

to allow unbounded and negative radius. The concept of formal ball was introduced by Weihrauch and Schreiber in 1981 and the negative radius case (the generalized formal ball) by Tsuiki and Hattori in 2008. Specifically, if

(X,d)

is a

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

and

\mathbb^

the nonnegative real numbers, then an element of

B^+(X,d)=X\times\mathbb^

is a formal ball. Elements of

B(X,d)=X\times\mathbb{R}

are known as generalized formal balls. Formal balls possess a partial order

\leq

defined by

(x,r)\leq(y,s)

d(x,y)\leq r-s

, identical to that defined by set inclusion. Generalized formal balls are interesting because this partial order works just as well for

B(X,d)

as for

B^+(X,d)

, even though a generalized formal ball with negative radius does not correspond to a subset of

X

. Formal balls possess the

Lawson topology In mathematics and theoretical computer science the Lawson topology, named after Jimmie D. Lawson, is a topology on partially ordered sets used in the study of domain theory. The lower topology on a poset ''P'' is generated by the subbasis consistin ...

and the Martin topology.

References

*K. Weihrauch and U. Schreiber 1981. "Embedding metric spaces into CPOs". ''Theoretical computer science'', 16:5-24. *H. Tsuiki and Y. Hattori 2008. "Lawson topology of the space of formal balls and the hyperbolic topology of a metric space". ''Theoretical computer science'', 405:198-205 *Y. Hattori 2010. "Order and topological structures of posets of the formal balls on metric spaces". ''Memoirs of the Faculty of Science and Engineering. Shimane University. Series B'' 43:13-26 Topology