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Boundary (topology)

In topology and mathematics in general, the **boundary** of a subset *S* of a topological space *X* is the set of points which can be approached both from *S* and from the outside of *S*. More precisely, it is the set of points in the closure of *S*, not belonging to the interior of *S*. An element of the boundary of *S* is called a **boundary point** of *S*. The term **boundary operation** refers to finding or taking the boundary of a set. Notations used for boundary of a set *S* include bd(*S*), fr(*S*), and ∂*S*. Some authors (for example Willard, in *General Topology*) use the term **frontier** instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology and manifold theory. However, frontier sometimes refers to a different set, which is the set of boundary points which are not actually in the set; that is, *S* \ *S*.

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