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Let $\Omega\colon \mathrm{Top}_*\to\mathrm{Top}_*$ be the loop space functor assigning to each pointed topological space $X$ the pointed space consisting of all based continuous maps $S^1\to X$ equipped with the compact-open topology.

Is there a topological characterisation of the objects lying in the range of $\Omega$?

I am aware of the results in terms of $A_\infty$-structures, but if I am not mistaken those results provide a homotopical characterisation, not a topological one. This implies that one must restrict attention to nice spaces such as CW complexes. I am interested instead in the category $\mathrm{Top}$.

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  • $\begingroup$ How could it possibly help you to know that some space is the loop space of the Hawaiian earring or something like that? $\endgroup$
    – Qiaochu Yuan
    Commented Dec 11, 2014 at 15:19
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    $\begingroup$ @QiaochuYuan I admit a sick interest in wild spaces. $\endgroup$
    – johndoe
    Commented Dec 11, 2014 at 15:22

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