Let $X$ be a topological space, and let $Y$ be the adjunction $Y := X\cup_\varphi D^k$ where $D^k$ is a closed $k$ ball and $\funcdef{\varphi}{S^{k-1}}{X}$ 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 $\varphi.$ The image $e^k$ of $D^k$ in $Y$ is called a closed $k$ cell, and the image $\oce^k$ of the interior $$ \ocD := D^k\setminus S^{k-1} $$ 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 $X\cup_\set{\varphi_i} D^{k_i}$ where the maps $\set{\varphi_i}$ into $X$ are defined on the boundary spheres of closed balls $\set{D^{k_i}}$