\begin{definition}[Definition:Möbius Strip/Formal Construction]
Let $T$ be the square embedded in the Cartesian plane defined as:
:$T = \closedint 0 1 \times \closedint 0 1$
Let $T'$ be the quotient space formed from $T$ using the identification mapping $p: T \to T'$ as follows:
:$\forall \tuple {x, y} \in T: \map p {x, y} = \begin {cases} \tuple {1, 1 - y} & : x = 0 \\ \paren {x, y} & : x \ne 0 \end {cases}$
Thus each point $\tuple {0, y}$ is mapped together with the point $\tuple {1, 1 - y}$ into the same point such that the set $\set 0 \times \closedint 0 1$ is mapped to $\set 1 \times \closedint 0 1$ but such that they are in the opposite direction.
\end{definition}