= Core circle of a Möbius band
{title2=$C=\{[u,0]:u\in[0,1]\}$}
The core circle is the image of the centre line in the rectangle defining a <Möbius band>. Its endpoints are identified, so it is a <circle>. The formula
$$
H_t([u,v])=[u,(1-t)v],\qquad 0\le t\le1,
$$
defines a <deformation retraction> onto that <circle>: it respects $(0,v)\sim(1,-v)$, starts at the <identity map>, and fixes the core throughout. Consequently the <Möbius band> has <fundamental group> isomorphic to $\mathbb Z$, just like a <circle>; this alone cannot distinguish it from an <annulus>.
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