These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. They also show that it is possible for the boundary of a subset to contain a non-empty open subset of ; that is, for the interior of in to be non-empty. However, a subset's boundary always has an empty interior. In the space of rational numbers with the usual topology (the subspace topology of ), the boundary of where is irrational, is empty. The boundary of a set is a topological notion and may change if one changes the topology. For example, given the usual topology on the boundary of a closed disk is the disk's surrounding circle: If the disk is viewed as a set in with its own usual topology, that is, then the boundary of the disk is the disk itself: If the disk is viewed as its own topological space (with the subspace topology of ), then the boundary of the disk is empty. Boundary of an open ball vs. its surrounding sphere This example demonstrates that the topological boundary of an open ball of radius is necessarily equal to the corresponding sphere of radius (centered at the same point); it also shows that the closure of an open ball of radius is necessarily equal to the closed ball of radius (again centered at the same point). Denote the usual Euclidean metric on by which induces on the usual Euclidean topology. Let denote the union of the -axis with the unit circle centered at the origin ; that is, which is a topological subspace of whose topology is equal to that induced by the (restriction of) the metric In particular, the sets and are all closed subsets of and thus also closed subsets of its subspace Henceforth, unless it clearly indicated otherwise, every open ball, closed ball, and sphere should be assumed to be centered at the origin and moreover, only the metric space will be considered (and not its superspace ); this being a path-connected and locally path-connected complete metric space. Denote the open ball of radius in by so that when then is the open sub-interval of the -axis strictly between and The unit sphere in ("unit" meaning that its radius is ) is while the closed unit ball in is the union of the open unit ball and the unit sphere centered at