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## homeomorphism related question, how to show two space is homeormophic to each other or not [closed]

Hello

I am having difficulties in solving the following two questions.

1) For the first question, the author of the text states that if f:[a,b]-->R is a map, then Im f is a closed, bounded interval.

Question: Let X bu subset of R, and X is the union of the open intervals (3n, 3n+1) and the points 3n+2, for n= 0,1,2,... Let Y=(X-{2})union {2}. Prove that there are continuous bijections f:X--->Y, g:Y--->X, but that X, Y are not homeomorphic.

I can create the bijection from X to Y, and Y to X.

From X to Y, I would map {2} to {1} and everything else gets map to itself, so I get both injective and surjective mapping. From the Y to X direction, i would just map {1} to {2} and everything else gets map to itself. I get again a bijective mapping But how do I show that map from X to Y and also Y to X are both continuous. X is composed of open intervals and singletons, likewise for the set Y. Am I suppose to impose some sort of topology on X and Y and then described a basis elements. Also, why is the set Im F and Im g not counded or closed?

For the second problem:

Construct the homeomorphism f:[0,1]x[0,1]--->[0,1]x[0,1] such that f maps [0,1]x{0,1} union {0}x[0,1] onto {0}x[0,1].

My difficulties with this question are several

first: Am i to interpret [0,1]x{0,1} union {0}x[0,1] to mean ([0,1]x{0,1]) union ({0}x[0,1]). If so then ([0,1]x{0,1}) union ({0}x[0,1]) is a subset of ({0,1] union (0})x({0,1} union [0,1]). By property of cartesian product. I am not sure how to proceed from here onwards.