Euclidean topology induced by the Euclidean metric). contains the origin and for every 1-dimensional vector subspace of is a neighborhood of the origin in when is given its unique Hausdorff vector topology. Connection to vector/TVS topologies: This condition gives insight as to why every neighborhood of the origin in every topological vector space (TVS) is necessarily absorbing: If is a neighborhood of the origin in a TVS then for every 1-dimensional vector subspace is a neighborhood of the origin in when is endowed with the subspace topology induced on it by This subspace topology is always a vector topology and because is 1-dimensional, the only vector topologies on it are the Hausdorff Euclidean topology and the trivial topology, which is a subset of the Euclidean topology. So regardless of which of these vector topologies is on the set will be a neighborhood of the origin in with respect to its unique Hausdorff vector topology (the Euclidean topology). Thus is absorbing. The reason why the Euclidean topology is distinguished in this characterization ultimately stems from the defining requirement on TVS topologies that scalar multiplication be continuous when the scalar field is given this (Euclidean) topology. contains the origin and for every 1-dimensional vector subspace of is absorbing in the Here "absorbing" means absorbing according to any defining condition other than this one. This characterization shows that the property of being absorbing in depends on how behaves with respect to 1 (or 0) dimensional vector subspaces of In contrast, if a finite-dimensional vector subspace of has dimension and is endowed with its unique Hausdorff TVS topology, then being absorbing in is no longer sufficient to guarantee that is a neighborhood of the origin in (although it will still be a necessary condition). For this to happen, it suffices for to be an absorbing set that is also convex balanced and closed in (such a set is called a and it will be a neighborhood of the origin in because like every finite-dimensional Euclidean space, is a barrelled space). If then to this list can be appended: The algebraic interior of contains the origin (that is, ). If is balanced then to this list can be appended: For every there exists a scalar such that (or equivalently, such that ). For every there exists a scalar such that If is convex or balanced then to this list can be appended: For every there