Manifold with Boundary

A manifold with boundary of dimension is a topological space that is a T₂-space with a second countable topology in which every point has a neighborhood homeomorphic to a relatively open set in the closed half-space

$H^n={(x_1,...,x_n) in R^n:x_n>=0}.$

Points represented with form the boundary, which is an -dimensional manifold without boundary. The closed ball is a manifold with boundary whose boundary is a sphere.

For , the same topological conditions apply with local model , and the manifold boundary is empty (Lee 2013, p. 27).

A manifold with boundary may have empty manifold boundary. It is then a manifold without boundary (Lee 2013, pp. 26-27).

See also

Boundary, Manifold, Manifold without Boundary, Open Manifold

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References

Lee, J. M. Introduction to Topological Manifolds, 2nd ed. New York: Springer, 2011.Lee, J. M. Introduction to Smooth Manifolds, 2nd ed. New York: Springer, 2013. https://doi.org/10.1007/978-1-4419-9982-5.

Cite this as:

Weisstein, Eric W. "Manifold with Boundary." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ManifoldwithBoundary.html

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