cell attachment

Let $X$ be a topological space, and let $Y$ be the adjunction $Y:=X{\cup }_{\phi }{D}^k$, where $D^k$ is a closed $k$-ball (http://planetmath.org/StandardNBall) and $\phi :S^{k-1}\to X$ is a continuous map, with $S^{k-1}$ is the $(k-1)$-sphere considered as the boundary of $D^k$. Then, we say that $Y$ is obtained from $X$ by the attachment of a $k$-cell, by the attaching map $\phi .$ The image $e^k$ of $D^k$ in $Y$ is called a closed $k$-cell, and the image ${\stackrel{\circ }{e}}^k$ of the interior

$$D^{\circ }:=D^k\setminus S^{k-1}$$

of $D^k$ is the corresponding open $k$-cell.

Note that for $k=0$ the above definition reduces to the statement that $Y$ is the disjoint union of $X$ with a one-point space.

More generally, we say that $Y$ is obtained from $X$ by cell attachment if $Y$ is homeomorphic to an adjunction $X{\cup }_{\left\{{\phi }_i\right\}}{D}^{k_i}$, where the maps $\left\{{\phi }_i\right\}$ into $X$ are defined on the boundary spheres of closed balls $\left\{D^{k_i}\right\}$.

Title	cell attachment
Canonical name	CellAttachment
Date of creation	2013-03-22 13:25:53
Last modified on	2013-03-22 13:25:53
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	13
Author	yark (2760)
Entry type	Definition
Classification	msc 54B15
Synonym	cell adjunction
Related topic	CWComplex
Defines	cell
Defines	open cell
Defines	closed cell
Defines	attaching map