The unit interval is the topological space $I = [0,1] \subset \mathbb{R}$ regarded as a subspace of the real line, with the standard Euclidean metric topology. It is used to define the fundamental groupoid $\Pi_1(X)$ of paths in a topological space $X$. A path in $X$ is simply a continuous function $p \colon I \to X$. The path has two endpoints $p(0), p(1)\in X$ defined by evaluating at the endpoints $0,1 \in I$. If $q \colon I \to X$ is a second path with the property that $p(1)=q(0)$, then there exists a composite path $p \ast q \colon I \to X$ defined by the composite continuous function \(\xymatrix{ I \ar[r]^-{\delta}_-\cong & I \vee I \ar[r]^-{p \vee q} & X.}\) Here $I \vee I$ is the space formed by gluing two copies of $I$ together by identifying the point $1$ in the left-hand copy with the point $0$ in the right-hand copy: \(\xymatrix{ \ast \ar[d]_{1} \ar[r]^{0} \ar@{}[dr]\mid(.8){\displaystyle\ulcorner} & I \ar[d] \\ I \ar[r] & I \vee I}\) The space $I \vee I$ is homeomorphic to the space $[0,2] \subset \mathbb{R}$. The map $\delta \colon I \to I \vee I$ is the homeomorphism $t \mapsto 2t$. Note that this map sends the endpoints of the domain $I$ to the endpoints of the fattened interval $I \vee I$. Thus, $(p \ast q)(0) = p(0)$ and $(p \ast q)(1) = q(1)$; that is, $p \ast q$ is a path in $X$ from the starting point of the path $p$ to the ending point of the path $q$.