Can a cell-complex have no zero cell?

Question

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.

What do you mean by "cell complex"? Are you using that as a synonym for CW-complex? — Eric Wofsey, Jan 02 '19 at 07:44
No, I mean a cellular complex, which is basically the same as a CW complex with no closure-finiteness or weak-topology condition (see link ) — TryingToGetOut, Jan 02 '19 at 07:49

score 6 · Accepted Answer · answered Jan 02 '19 at 07:54

6

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!)

answered Jan 02 '19 at 07:54

Eric Wofsey

342,377

1

thanks, this seems to answer my question accurately! But is it necessary in the definition that the n-cells have attaching maps to the (n-1)-skeleton? Because the 0-cell will not have that for instance. – TryingToGetOut Jan 02 '19 at 07:58
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In this game, the boundary of the $0$-disc is empty so it attaches perfectly well to the empty $-1$-skeleton :-) @TryingToGetOut – Angina Seng Jan 02 '19 at 08:01
Yes of course.. Thanks! – TryingToGetOut Jan 02 '19 at 08:07

Can a cell-complex have no zero cell?

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