Contrasting Terms
A compact manifold means a "manifold" that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary (the boundary may be empty). By contrast, a closed manifold is compact without boundary.
An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.
The notion of closed manifold is unrelated with that of a closed set. A disk with its boundary is a closed set, but not a closed manifold.
Read more about this topic: Closed Manifold
Famous quotes containing the words contrasting and/or terms:
“We could not help contrasting the equanimity of Nature with the bustle and impatience of man. His words and actions presume always a crisis near at hand, but she is forever silent and unpretending.”
—Henry David Thoreau (1817–1862)
“Women have acquired equal place to man in society, but the double standard has really never been relinquished; certainly not by men. Modern man’s fear of passivity or of the active woman proves to be as eternal as modern woman’s struggle to come to terms with her femininity.”
—Peter Blos (20th century)