Connectivity and closedness combine to make surface property {genus, surface}|. Closed Riemann surfaces with same genus are topologically equivalent. Genus is invariant under birational transformation.
sphere
Sphere has genus 0. Genus-0 closed surfaces can map onto spheres and connect simply. Sphere projective planes have genus zero, are closed, and look like circles with semi-circumference line at infinity.
handles
Sphere with n handles has genus n. The one-sided-surface Klein's bottle has genus one, because it has one handle.
holes
Sphere with n holes has genus n. Hole number equals function branch-point number divided by two, minus function-value number, plus one. For curves, genus equals 0.5 * (n - 1) * (n - 2) - d, where d equals double-point number and n equals function degree. Riemann surface corresponding to genus-p curve has connectivity 2*p + 1.
Mathematical Sciences>Topology>Surface>Genus
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Date Modified: 2022.0224