# A problem in Wedge product of topological spaces

I am studying algebraic topology and I am a beginner in it. I want to prove that if $$(X, x_0) \sim (Y, y_0)$$ and $$(Z, z_0) \sim (W , w_0)$$ then $$X\vee Z \sim Y\vee W .$$ X,Y, Z, W are top spaces

I have just an intuition that wedge product seems something like glueing two spaces together in a point. I don't know how can I start to think about it and solve it. Any help would be great thanks.

• This is most easily done by mapping properties of wedges and quotients. – Randall Jun 21 '19 at 11:40

• You start by assuming that $$(X, x_0) \sim (Y, y_0)$$ and $$(Z, z_0) \sim (W , w_0)$$. This is extremely important. That tells you that you have some maps back and forth which satisfy a certain property to do with their composition being homotopy equivalent to the identity. Keep these maps, and the homotopies which give you your homotopy equivalences, in mind.
• You can use these maps back and forth to build yourself some maps back and forth on the wedges of spaces $$X\vee Z \leftrightarrow Y\vee W$$.