In the subject of
manifold theory in
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, if
is a topological
manifold with boundary
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, its double is obtained by gluing two copies of
together along their common boundary. Precisely, the double is
where
for all
.
If
has a smooth structure, then its double can be endowed with a smooth structure thanks to a
collar neighbourdhood.
Although the concept makes sense for any manifold, and even for some non-manifold sets such as the
Alexander horned sphere
The Alexander horned sphere is a pathological object in topology discovered by . It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topological 3-ball, the Alexande ...
, the notion of double tends to be used primarily in the context that
is non-empty and
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
.
Doubles bound
Given a manifold
, the double of
is the boundary of