Wednesday, 5 December 2018

real analysis - Is there a continuous function from X onto Y?

(1) X=(0,1] and Y=[0,1];



(2) X=[0,1] and Y is the topologist's sine curve



(3) X=[0,1][2,3] and Y=X×X




I believe there is a continuous function for the first one.



I know there doesn't exist a continuous function for the second one, because X is path connected and the topologist's sine curve is not. As well as, f(0) doesn't exist in Y.



I feel like there is no continuous function for (3) because of the disconnection; however, I could be wrong.

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