Component of constant loop in loop space of based topological space

Definition

Definition as a based topological space

Suppose $(X,x_{0})$ is a based topological space. The component of constant loop in loop space of based topological space for $(X,x_{0})$, denoted $\Omega _{0}(X,x_{0})$ or $\Omega _{0}^{1}(X,x_{0})$, is defined as the path component of the constant loop at $x_{0}$ in the loop space of a based topological space $\Omega (X,x_{0})$ (which is equipped with the subspace topology from the compact-open topology on the space of all continuous maps). The basepoint is chosen as the constant loop at $x_{0}$.

At least in the case where $X$ is a compactly generated Hausdorff space, this is the same as the space of nullhomotopic loops based at $x_{0}$, because, under these assumptions, paths between loops in $\Omega (X,x_{0})$ are equivalent to homotopies of loops.

Definition as a H-space

All this is true at least in the case where $X$ is a compactly generated Hausdorff space.

The space $\Omega _{0}(X,x_{0})$ can be given the structure of a topological magma, and in fact, a H-space, by defining the composition of two loops by concatenation, or equivalently, restricting the natural H-space structure from $\Omega (X,x_{0})$. In fact, $\Omega _{0}(X,x_{0})$ is the inverse image of the identity element in $\pi _{1}(X,x_{0})$ under the homomorphism:

$\!\Omega (X,x_{0})\to \pi _{0}(\Omega (X,x_{0}))=\pi _{1}(X,x_{0})$