nLab
boundary

Contents

Definition

Of a subset of a topological space

For SXS \subset X a subset of a topological space XX, the boundary or frontier S\partial S of SS is its closure S¯\bar S minus its interior S S^\circ:

S=S¯\S \partial S = \bar S \backslash S^\circ

Of a manifold

In a manifold with boundary of dimension nn the boundary is the collection of points which do not have a neighborhood diffeomorphic to an open n-ball, but do have a neighborhood homeomorphic to a half-ball, that is, an open ball intersected with closed half-space.

H n=Z n/B nH_n = Z_n/B_n(chain-)homology(cochain-)cohomologyH n=Z n/B nH^n = Z^n/B^n
C nC_nchaincochainC nC^n
Z nC nZ_n \subset C_ncyclecocycleZ nC nZ^n \subset C^n
B nC nB_n \subset C_nboundarycoboundaryB nC nB^n \subset C^n

Revised on April 27, 2013 13:04:38 by Urs Schreiber (89.204.139.246)