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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).
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 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]
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 ...
A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior. [21] An example of a non-regular open set is the set U = (0,1) ∪ (1,2) in R with its normal topology, since 1 is in the interior of the closure of U , but not in U .
Example: the blue circle represents the set of points (x, y) satisfying x 2 + y 2 = r 2.The red disk represents the set of points (x, y) satisfying x 2 + y 2 < r 2.The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.
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.