fact is one of the primary motivations for even defining the property "absorbing in " If is a disk (convex and balanced) then so that in particular, is an absorbing subset of Thus if is a disk in then is absorbing in if and only if This conclusion is not guaranteed if the set is balanced but not convex; for example, the union of the and axes in is a non-convex balanced set that is not absorbing in Any superset of an absorbing set is absorbing. Thus the union of any family of (one or more) absorbing sets is absorbing. The image of an absorbing set under a surjective linear operator is again absorbing. The inverse image of an absorbing subset (of the codomain) under a linear operator is again absorbing (in the domain). The intersection of a finite family of (one or more) absorbing sets is absorbing. If absorbing then the same is true of the symmetric set Auxiliary normed spaces If is convex and absorbing in then the symmetric set will be convex and balanced (also known as an or a ) in addition to being absorbing in This guarantees that the Minkowski functional of will be a seminorm on thereby making into a seminormed space that carries its canonical pseduometrizable topology. The set of scalar multiples as ranges over (or over any other set of non-zero scalars having as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If is a topological vector space and if this convex absorbing subset is also a bounded subset of then the same will be true of the absorbing disk in which case will be a norm and will form what is known as an auxiliary normed space. If this normed space is a Banach space then is called a . Properties Every absorbing set contains the origin. If is an absorbing disk in a vector space then there exists an absorbing disk in such that If is an absorbing subset of then and more generally, for any sequence of scalars such that Consequently, if a topological vector space is a non-meager subset of itself (or equivalently for TVSs, if it is a Baire space) and if is a closed absorbing subset of then necessarily contains a non-empty open subset of (in other words, 's topological interior will not be empty), which