# Cellular mapping

A mapping $f: ( X, A) \rightarrow ( Y, B)$ from one relative CW-complex $( X, A)$ into another $( Y, B)$ such that $f (( X, A) ^ { p } ) \subset ( Y, B) ^ { p }$, where $( X, A) ^ { p }$ and $( Y, B) ^ { p }$ are the $p$- skeletons of $X$ and $Y$ relative to $A$ and $B$, respectively. In the case when $A, B = \emptyset$, one obtains a cellular mapping $f$ from the CW-complex $X$ into the CW-complex $Y$.

A homotopy $F: ( X, A) \rightarrow I \times ( Y, B)$, where $I = [ 0, 1]$, is called cellular if $F (( X, A) \times I) ^ { p } \subset ( Y, B) ^ { p }$ for all $p$. The following cellular approximation theorem is the analogue of the simplicial approximation theorem (see Simplicial mapping): Let $f: ( X, A) \rightarrow ( Y, B)$ be a mapping from one relative CW-complex $( X, A)$ into another $( Y, B)$ the restriction of which to some subcomplex $( L, N) \subset ( X, A)$ is cellular. Then there exists a cellular mapping $g: ( X, A) \rightarrow ( Y, B)$ that is homotopic to $f$ relative to $L$.