# Cell complex

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A separable space $X$ that is a union of non-intersecting cells. Here, by a $p$- dimensional cell one means a topological space that is homeomorphic to the interior of the unit cube of dimension $p$. If for each $p$- dimensional cell $t ^ {p}$ of $X$ one is given a continuous mapping $f$ from the $p$- dimensional cube $I ^ {p}$ into $X$ such that: 1) the restriction $f ^ { * }$ of $f$ to the interior $\mathop{\rm Int} I ^ {p}$ of $I ^ {p}$ is one-to-one and the image $f ( I ^ {p} )$ is the closure $\overline{ {t ^ {p} }}\;$ in $X$ of $t ^ {p}$( here $f ^ { * }$ is a homeomorphism of $\mathop{\rm Int} I ^ {p}$ onto $t ^ {p}$); and 2) the set $f ( \partial I ^ {p} )$, where $\partial I ^ {p}$ is the boundary of $I ^ {p}$, is contained in the union $X ^ {p-} 1$ of the cells $t ^ {p-} 1$ of $X$, then $X$ is called a cell complex; the union $X ^ {p-} 1$ is called the skeleton of dimension $p - 1$ of the cell complex $X$. An example of a cell complex is a simplicial polyhedron.
A subset $L$ of a cell complex $X$ is called a subcomplex if it is a union of cells of $X$ containing the closures of such cells. Thus, the $n$- dimensional skeleton $X ^ {n}$ of $X$ is a subcomplex of $X$. Any union and any intersection of subcomplexes of $X$ are subcomplexes of $X$.