环路空间的局部性质：寻求半局部性质对应结论的参考文献
We all know the foundational isomorphism in homotopy theory:
$$\pi_k(X)=\pi_{k-1}(\Omega X)$$
A key special case here is that $\Omega X$ is path-connected if and only if $X$ is simply-connected.
Now, regarding the corresponding (semi-)local properties you're inquiring about—$\Omega X$ is (semi-)locally path-connected if and only if $X$ is (semi-)locally simply-connected—here are some solid references to consult:
Topology: An Introduction by Klaus Jänich: This textbook ties basic homotopy theory to loop space properties nicely. Flip to chapters covering covering spaces and loop spaces; since (semi-)local simple connectedness is deeply linked to covering space theory, you’ll find explicit connections to the (semi-)local path connectedness of $\Omega X$ here.
Algebraic Topology by Allen Hatcher: While Hatcher doesn’t state this equivalence in a single explicit line, you can piece together the logic from his sections on loop spaces, covering spaces, and local properties. The "Fundamental Group and Covering Spaces" chapter breaks down (semi-)local simple connectedness, and the loop space content in "Homotopy Groups" gives you the tools to connect this to $\Omega X$’s local path connectivity.
Homotopy Theory: An Introduction to Algebraic Topology by William S. Massey: This book includes more granular discussions on the relationship between loop spaces and local connectivity traits. Check the sections dedicated to loop spaces and local simple connectedness—Massey frequently draws clear, direct lines between these homotopy-theoretic properties.
If you want to work through the proof independently, the core intuition is this: (semi-)local simple connectedness of $X$ means every point has neighborhoods where all loops are nullhomotopic. This translates directly to neighborhoods in $\Omega X$ (made up of loops that stay within those $X$-neighborhoods) where any two loops can be connected by a homotopy—exactly the definition of (semi-)local path connectedness for $\Omega X$.
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