Definition. Let be a topological space and let be a subset of . The boundary (or frontier) of is the set , where the overline denotes the closure of a set. Instead of , many authors use some other notation such as , , or . Note that the symbol is also used for other meanings of `boundary'.

From the definition, it follows that the boundary of any set is a closed set. It also follows that , and .

The term `boundary' (but not `frontier') is used in a different sense for topological manifolds: the boundary of a topological -manifold is the set of points in that do not have a neighbourhood homeomorphic to . (Some authors define topological manifolds in such a way that they necessarily have empty boundary.) For example, the boundary of the topological -manifold is .