Construction of the Möbius Strip
Consider the unit square $Z = [0, 1] \times [0, 1]$ in $\mathbb{R}^2$ and let $X = \{ 0 \} \times [0, 1]$ and $Y = \{ 1 \} \times [0, 1]$ as illustrated below:
Define a function $f : X \to Y$ by $f(0, x) = (1, 1 - x)$.
The resulting space $Z_f$ is the extremely famous Möbius Strip.