= Gluing annuli with one reflected boundary
Glue the outer circles of two closed <annuli> by the identity, and their inner circles by complex conjugation. The result is a <Klein bottle>. To see the gluing explicitly, let $r\in[1/2,1]$ and $z\in S^1$. Parametrize the first annulus by $(z,\tau)=(z,2r-1)$, with $0\leq\tau\leq1$, and the second by $(z,\tau)=(z,3-2r)$, with $1\leq\tau\leq2$. The outer-circle gluing identifies their common level $\tau=1$. The remaining gluing is
$$
(z,0)\sim(\overline z,2).
$$
This is the <mapping torus> of reflection of the <circle>. The coordinate maps and their piecewise inverses agree on every identified boundary, so the <universal property of the quotient topology> gives inverse continuous maps between the two quotient descriptions. Gluing collars by the reflection supplies disc charts at the seam; thus it is a closed <surface>. Cutting the circle coordinate at one point gives a rectangle with one matching and one reversed edge pairing, the usual Klein-bottle model. Its <Euler characteristic> is zero and it is a <nonorientable surface>, so the classification theorem identifies it with the connected sum of two real projective planes.
\Image[/past-exam-of-the-topology-course-of-the-university-of-oxford/2017-reflected-annulus-gluing.png]
{title=Annulus coordinates exhibit the Klein bottle as a reflection mapping torus}
{description=The identity outer-circle gluing forms the middle level. The two inner boundaries become the reflected cylinder ends; the vertical sides represent the periodic circle coordinate.}
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