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Topology:Exhibit spaces such that

Amer

Active member
Mar 1, 2012
275
Exhibits spaces X,Y and Z such that
[tex]X\; x\; Y [/tex] is homeomorphic to [tex]X \; x \; Z [/tex] but Y is not homeomorphic to Z
 

PaulRS

Member
Jan 26, 2012
37
Consider $X = Y \times Z \times Y \times Z \times ... $ , $Y = \{0,1\}$ and $Z = \{0\}$. Here $Y$ and $Z$ with the discrete topology and $X$ with the product topology.

Define $f : X \times Y \to X \times Z$ by $f \left( \left( x_1, x_2, x_3, ...\right) , y \right) = \left( \left( y, x_4, x_1,x_6,x_3,x_8,x_5,x_{10} ...\right) , x_2 \right)$

;)
 

Amer

Active member
Mar 1, 2012
275
nice example :)
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Consider $X = Y \times Z \times Y \times Z \times ... $ , $Y = \{0,1\}$ and $Z = \{0\}$. Here $Y$ and $Z$ with the discrete topology and $X$ with the product topology.

Define $f : X \times Y \to X \times Z$ by $f \left( \left( x_1, x_2, x_3, ...\right) , y \right) = \left( \left( y, x_4, x_1,x_6,x_3,x_8,x_5,x_{10} ...\right) , x_2 \right)$

;)
That construction actually proves a stronger result, namely:
Given any two topological spaces $Y$ and $Z$, there exists a space $X$ such that $X\times Y$ is homeomorphic to $X\times Z.$​