with the origin removed, this condition may be restated as: for some The non-strict inequality can be replaced with the strict inequality which is the next characterization. There exists a real such that for every non-zero scalar satisfying Or stated more succinctly, for some Here is the open ball with the origin removed and If is a balanced set then this list can be extended to include: There exists a non-zero scalar such that If then the requirement may be dropped. There exists a non-zero scalar such that If (a necessary condition for to be an absorbing set, or to be a neighborhood of the origin in a topology) then this list can be extended to include: There exists such that for every scalar satisfying Or stated more succinctly, There exists such that for every scalar satisfying Or stated more succinctly, This set inclusion is equivalent to (since ). Because this may be rewritten which gives the next statement. There exists such that There exists such that There exists such that The next characterizations follow from those above and the fact that for every scalar the balanced hull of satisfies and (since ) its balanced core satisfies There exists such that In words, a set is absorbed by if it is contained in some positive scalar multiple of the balanced core of There exists such that There exists a scalar such that In words, can be scaled to contain the balanced hull of There exists a scalar such that There exists a scalar such that In words, can be scaled so that its balanced core contains There exists a scalar such that There exists a non-zero scalar such that In words, the balanced core of contains some non-zero scalar multiple of If or this list can be extended to include: absorbs (according to any defining condition of "absorbs" other than this one). In other words, may be replaced by in the characterizations above if (or trivially, if ). A set absorbing a point A set is said to if it absorbs the singleton set A set absorbs the origin if and only if it contains the origin; that is, if and only if As detailed below, a set is said to be if it absorbs every point of This notion of one set absorbing another is also used in other definitions: A subset of a topological vector space is called