My question is very simple, but I wasn't able to find an answer in various sources. Cell-complexes are commonly presented using an inductive construction where $n$-cells are attached to $(n-1)$-cells, starting with the data of a collection $X^0$ of $0$-cells.
But sometimes, one will define a $2$-cell for instance, as being a sub-cell complex of something, raising the question: can a cell complex $0$-skeleton be empty? And more generally, can a cell-complex have empty skeletons until a dimension $k$?
It doesn't seem absurd to me that the answer should be yes, but it worries me to never see the case taken into account in the inductive construction.
Best Answer
Suppose $X$ is a nonempty cell complex and let $n$ be minimal such that $X$ has an $n$-cell. If $n>0$, then this $n$-cell has an attaching map $S^{n-1}\to X^{n-1}$ where $X^{n-1}$ is the $(n-1)$-skeleton of $X$. But by minimality of $n$, $X^{n-1}=\emptyset$. Since $S^{n-1}$ is nonempty, there are no maps $S^{n-1}\to\emptyset$, so this is a contradiction.
So, if $X$ is any nonempty CW-complex, it must have a $0$-cell. (Of course, the empty space is a CW-complex with no cells at all!)