Connected space please check if the proof is correct?

Let X,Y be topo spaces. Consider Z = X x Y and the product topology genearted by the projections px, py. Let A be a subset of Z. Suppose that X,Y are locally path connected. Show that A is connected if and only if A is path connected.





For let x be a point of X, let H be the set of points x’ of X which can be joined to x by a path in X, and let K = X - H. I assert that both H and K are open in X. This will imply that K is empty, since X is connected, and so that X is pathwise-connected.


First, we show that H is open. For if x’ is a point of H there exist an open neighborhood U of x’ such that for any point x” of U there exists a path in U from x’ to x”. But since x’ is in H there is also a path in X from x to x’. Juxtaposing these two paths we obtain a path in X from x to x”, so that x” in H.


Thus U c H and so H is open. (c = proper subset)


Second, we show that K is open. For if x’ is a point of K there exists an open neighborhood U of x’ such that for any point x” of U there exists a path in U


from x” to x’. Then x” is not in H since, otherwise, there would exist a path in


X from x to x” and hence, by juxtaposition, a path in X from x to x’, contrary to


the assumption that x’ in K. So x” in K, thus U c K and so K is open

Connected space please check if the proof is correct?
OK, you showed that a connected, locally path connected space is path connected. That is NOT what you were aksed to prove, though. You are being asked to show that a subset of the product is connected if and only if it is path connected.





It turns out that the statement is false. The problem is that the subset need not be locally path connected. As an example, take A to be the topologists sine curve as a subset of RxR.
Reply:I didn't see any mention of Y.





What you proved is that if a subset of X is connected, it's path connected.