Example

Let $I$ be the interval $[0, 1]$ in $\mathbf{R}$. If $Y$ is any topological space, the space $C_c(I; Y)$ is called the space of paths in $Y$. For each $y \in Y$, the subspace $\Omega_y(Y)$ of $C_c(I; Y)$ consisting of paths $u$ such that $u(0) = u(1) = y$ is called the space of loops (in $Y$) at the point $y$.