Is the Finite Cartesian Product of Simply Connected Sets Also Simply Connected?

[jimisrv]

Hi,

Easy question about connectedness. I have from Munkres that the finite Cartesian product of connected sets is connected. How about for simply connected sets? This seems like a natural extension of the theorem. Would you say I need to offer a proof? Simply connectedness seems to require some algebraic topology that I am not really familiar with.

Thanks,
Mike

 

[quasar987]

If you haven't seen the notion of simple connectedness, and are not expected to have seen it in the context in which your problem arises, then it probably means that there is another way to go.

But the fact that the product of two simply connected space is again simply connected seems easy to prove to me. You must show that path connectedness and triviality of the fundamental group is preserved by taking the product.