Consider the sets X={0} {1/n: nN} Y = {1, 1} U {1-1/n: nN} U{-1+1/n: nN}. with...

 

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Consider the sets X={0} {1/n: nN} Y = {1, 1} U {1-1/n: nN} U{-1+1/n: nN}. with the standard Euclidean metric on R. Show that X and Y are not homeomorphic, i.e. there does not exist a bijection f : XY which is continuous and f-1 is continuous as well. [5 marks] Note: Since X and Y are both countable, there exists a bijection f : XY. The claim above is that such f cannot be a homeomorphism. Consider the sets X={0} {1/n: nN} Y = {1, 1} U {1-1/n: nN} U{-1+1/n: nN}. with the standard Euclidean metric on R. Show that X and Y are not homeomorphic, i.e. there does not exist a bijection f : XY which is continuous and f-1 is continuous as well. [5 marks] Note: Since X and Y are both countable, there exists a bijection f : XY. The claim above is that such f cannot be a homeomorphism.