if it is absorbed by every neighborhood of the origin. A set is called if it absorbs every bounded subset. Examples Every set absorbs the empty set but the empty set does not absorbs any non-empty set. The singleton set containing the origin is the one and only singleton subset that absorbs itself. Suppose that is equal to either or If is the unit circle (centered at the origin ) together with the origin, then is the one and only non-empty set that absorbs. Moreover, there does exist non-empty subset of that is absorbed by the unit circle In contrast, every neighborhood of the origin absorbs every bounded subset of (and so in particular, absorbs every singleton subset/point). Absorbing set A subset of a vector space over a field is called an of and is said to be if it satisfies any of the following equivalent conditions (here ordered so that each condition is an easy consequence of the previous one, starting with the definition): Definition: absorbs every point of that is, for every absorbs So in particular, can not be absorbing if Every absorbing set must contain the origin. absorbs every finite subset of For every there exists a real such that for any scalar satisfying For every there exists a real such that for any scalar satisfying For every there exists a real such that Here is the open ball of radius in the scalar field centered at the origin and The closed ball can be used in place of the open ball. Because the inclusion holds if and only if This proves the next statement. For every there exists a real such that where Connection to topology: If is given its usual Hausdorff Euclidean topology then the set is a neighborhood of the origin in thus, there exists a real such that if and only if is a neighborhood of the origin in Consequently, satisfies this condition if and only if for every is a neighborhood of in when is given the Euclidean topology. This gives the next characterization. The only TVS topologies on a 1-dimensional vector space are the (non-Hausdorff) trivial topology and the Hausdorff Euclidean topology. Every 1-dimensional vector subspace of is of the form for some and if this 1-dimensional space is endowed with the (unique) , then the map defined by is necessarily a TVS-isomorphism (where as usual, is endowed with its standard