Boundary (algebraic 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.
A connected component of the boundary of S is called a boundary component of S.
If the set consists of discrete points only, then the set has only a boundary and no interior.
There are several common (and equivalent) definitions to the boundary of a subset S of a topological space X:
Consider the real line R with the usual topology (i.e. the topology whose basis sets are open intervals). One has
These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.
In the space of rational numbers with the usual topology (the subspace topology of R), the boundary of , where a is irrational, is empty.
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