Crumpled cube
Encyclopedia
In geometric topology
, a branch of mathematics, a crumpled cube is a 2-sphere
in R3
together with its interior
. showed that the union
of a crumpled cube and a 3-ball
glued along their boundaries
is a 3-sphere.
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
, a branch of mathematics, a crumpled cube is a 2-sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
in R3
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
together with its interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
. showed that the union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of a crumpled cube and a 3-ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
glued along their boundaries
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...
is a 3-sphere.