In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold.
The simplest example is a circle, which is a compact one-dimensional manifold. Other examples of closed manifolds are the torus and the Klein bottle. As a counterexample, the real line is not a closed manifold because it is not compact. A disk is a compact two-dimensional manifold, but is not a closed manifold because it has a boundary.
Compact manifolds are, in an intuitive sense, "finite". By the basic properties of compactness, a closed manifold is the disjoint union of a finite number of connected closed manifolds. One of the most basic objectives of geometric topology is to understand what the supply of possible closed manifolds is.
All compact topological manifolds can be embedded into for some n, by the Whitney embedding theorem.
Read more about Closed Manifold: Contrasting Terms, Use in Physics
Famous quotes containing the words closed and/or manifold:
“Don: Why are they closed? They’re all closed, every one of them.
Pawnbroker: Sure they are. It’s Yom Kippur.
Don: It’s what?
Pawnbroker: It’s Yom Kippur, a Jewish holiday.
Don: It is? So what about Kelly’s and Gallagher’s?
Pawnbroker: They’re closed, too. We’ve got an agreement. They keep closed on Yom Kippur and we don’t open on St. Patrick’s.”
—Billy Wilder (b. 1906)
“Thy love is such I can no way repay,
The heavens reward thee manifold I pray.
Then while we live, in love lets so persever,
That when we live no more, we may live ever.”
—Anne Bradstreet (c. 1612–1672)