I'm in trouble with CW complexes. I want to know, how to prove that the wedge sum $S^1\vee S^1$ of $(S^1,x_0)$ and $(S^1,y_0)$ is a finite CW complex, $y_0$ and $x_0$ are base points
My explanation is that you have the two 0-cells $x_0,y_0$ and two 1 cells, a line, and by gluing this cells you obtain the two $S^1$. But I don't know how to continue, how to argue with disjoint union and then passing to the quotient.
The finiteness is clear.
Regards
Let us say $X$ is the CW complex obtained by gluing together one $0$-cell and two $1$-cells, with $q$ being the canonical quotient map from the disjoint union of the balls $D^n_\alpha$ to the complex $X$. Let $q_0,q_1$ be the quotient maps which correspond to the circles being constructed as CW complexes, and let $w$ be the quotient map from $S^1\sqcup S^1$ to $S^1\vee S^1$ which identifies the two $0$-cells. If $f$ is the map sending both balls $D^0_0$ and $D^0_1$ to a single ball $D^0$, can you show that $qf$ induces a homeomorphism $h:S^1\vee S^1 \to X$ ?