# Topology:Exhibit spaces such that

#### Amer

##### Active member
Exhibits spaces X,Y and Z such that
$$X\; x\; Y$$ is homeomorphic to $$X \; x \; Z$$ but Y is not homeomorphic to Z

#### PaulRS

##### Member
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 $f \left( \left( x_1, x_2, x_3, ...\right) , y \right) = \left( \left( y, x_4, x_1,x_6,x_3,x_8,x_5,x_{10} ...\right) , x_2 \right)$

nice example

#### Opalg

##### MHB Oldtimer
Staff member
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 $X\times Z.$​