# Every CW complex is homotopy equivalent to a CW complex with 1 0-cell

I want to prove that Every path connected CW complex $$X$$ is homotopy equivalent to a complex $$Y$$ with a single $$0$$-cell. Notice that I'm making no assumption on the dimension of $$X$$ or the number of cells.

My idea would be to take the $$0$$-cells in $$X$$ and collapse them to a point. So, I consider $$X^{(1)}$$ (the $$1$$-skeleton) and a maximal tree $$T$$ (maximal among the contractable $$1$$-dimensional subcomplexes of $$X^{(1)}$$) in $$X^{(1)}$$, then $$T$$ is a subcomplex of $$X^{(1)}$$ which is a subcomplex of $$X$$, so $$T$$ is a subcomplex of $$X$$. Moreover, $$T$$ is contractable and contains every $$0$$-cell of $$X^{(1)}$$ (so every $$0$$-cell of $$X$$). We conclude $$X/T$$ is a CW complex with a single $$0$$-cell and the projection $$X\rightarrow X/T$$ is a homotopy equivalence. The existence of a maximal tree is proved with Zorn's lemma.

Is there some problem with this argument?

There is a way to construct a CW-approximation $$Z\to X$$ for any path-connected space $$X$$ such that $$Z$$ only has a single $$0$$-cell. This construction can be seen in Hatcher's book in the section on CW-approximations. Then if $$X$$ is a CW-complex, it follows from Whitehead's Theorem that the approximation is a homotopy equivalence.