  We investigate the classification of topological quandles on some simple
manifolds. Precisely we classify all Alexander quandle structures, up to
isomorphism, on the real line and the unit circle. For the closed unit interval
$[0, 1]$, we conjecture that there exists only one topological quandle
structure on it, i.e. the trivial one. Some evidences are provided to support
our conjecture.
