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There seems no reason to believe it is in general, for the only thing we know is that if two spaces X, Y are path connected, then they are connected, and their product X x Y is connected.

So I thought of this Lemma:

Let X, Y be two path connected spaces. Their product X x Y is path connected if the domains of the paths fx in X and fy in Y coincide.

Assume the hypothesis is true, then for X and Y there exist continuous functions fx : [a, b] --> X and fy : [a, b] --> Y, such that, for any pair of points x1, x2 of X and y1, y2 of Y fx(a) = x1, fx(b) = x2 and fy(a) = y1, fy(b) = y2.

Let f : [a, b] --> X x Y be given with f(x) = (fx(x), fy(x)). Since fx and fy are continuous, f is continuous too, and for any pair of points (x1, x2), (y1, y2) of X x Y f(a) = (x1, y1), f(b) = (x2, y2).

I think this should work, any comments?