If X and Y are path connected, show that XxY is also path connected.

For a fixed and

Let A be the set of all the points that join by a path in .

Since X and Y are both path connected, there exist such that , where x is in and y is in , contained in open sets and , respectively.

Let then a path can be connected from to x to .

Similarly for y

How can now show XxY is path connected?

How can I translate this to the product topology?

Can I say:

. Therefore the exist a path from to to ????