nLab boundary

Context

Topology

topology

algebraic topology

Contents

Definition

Of a subset of a topological space

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

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

Of a manifold

In a manifold with boundary of dimension $n$ 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.

$H_n = Z_n/B_n$(chain-)homology(cochain-)cohomology$H^n = Z^n/B^n$
$C_n$chaincochain$C^n$
$Z_n \subset C_n$cyclecocycle$Z^n \subset C^n$
$B_n \subset C_n$boundarycoboundary$B^n \subset C^n$

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