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 ????