Component of constant loop in loop space of based topological space

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 $$\Omega^1_0(X,x_0)$$, is defined as the path component of the constant loop at $$x_0$$ in the defining ingredient::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:

$$\! \Omega(X,x_0) \to \pi_0(\Omega(X,x_0)) = \pi_1(X,x_0)$$