I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is:
If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 and X2 is homeomorphic to the product of Y1 and Y2), to prove is that the components might not be homeomorphic. There is a hint: Let's consider $X_{1}=X_{2}=Y_{1}=\mathbb{N}$ and $Y_{2}= \left \{ p \right \}$ with the discrete topology.
Okay, we know from the definition that in discrete topology all sets are open, this means that {p} is open too...I don't understand how to prove that the components might be not homeomorphic...can somebody explain me?
Thanks in advance.