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In this sense interior and closure are dual notions. The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).
A set (in light blue) and its boundary (in dark blue). In topology and mathematics in general, the boundary of a subset S of a topological space X 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.
If is a closed submanifold-with-boundary of a manifold , then the relative interior (that is, interior as a manifold) of is locally closed in and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset. [2]
Each clopen subset of (which includes and itself) is simultaneously a regular open subset and regular closed subset. The interior of a closed subset of is a regular open subset of and likewise, the closure of an open subset of is a regular closed subset of . [2] The intersection (but not necessarily the union) of two regular open sets is a ...
A smooth 2-manifold: The interior chart with transition map φ 1 maps an open subset around an interior point to an open Euclidean subset, while the boundary chart with transition map φ 2 maps a closed subset around a boundary point to a closed Euclidean subset. The boundary is itself a 1-manifold without boundary, so the chart with transition ...
If is a topological space and is a point in , then a neighbourhood [1] of is a subset of that includes an open set containing , . This is equivalent to the point p ∈ X {\displaystyle p\in X} belonging to the topological interior of V {\displaystyle V} in X . {\displaystyle X.}
Suppose M is a 3-manifold with boundary. Suppose also that S is a compact surface with boundary that is properly embedded in M, meaning that the boundary of S is a subset of the boundary of M and the interior points of S are a subset of the interior points of M.
If is a subset of a topological space then the limit of a convergent sequence in does not necessarily belong to , however it is always an adherent point of . Let ( x n ) n ∈ N {\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }} be such a sequence and let x {\displaystyle x} be its limit.