The Product of Topological Spaces Does Not Obey Cancellation

Exercise 1. Find metric topological spaces [tex]A,B,C[/tex] such that [tex]A[/tex] is not homeomorphic to [tex]B[/tex], but [tex]A\times C[/tex] is homeomorphic to [tex]B\times C[/tex].

Exercise 2. Find path connected metric topological spaces [tex]A,B,C[/tex] such that [tex]A[/tex] is not homeomorphic to [tex]B[/tex], but [tex]A\times C[/tex] is homeomorphic to [tex]B\times C[/tex].

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