Editing Topology (section)

== <span style="color: #FFFFFF;">Analyzing</span> ==
{| class="wikitable"
|+ Topological Invariants of Surfaces
! Surface !! χ (Euler char) !! π₁ (fundamental group) !! Genus !! Orientable?
|-
| Sphere S² || 2 || Trivial {e} || 0 || Yes
|-
| Torus T² || 0 || ℤ × ℤ || 1 || Yes
|-
| Double torus || −2 || Complex (genus 2 surface group) || 2 || Yes
|-
| Real projective plane ℝP² || 1 || ℤ/2ℤ || — || No
|-
| Klein bottle || 0 || Non-abelian extension || — || No
|}

'''Famous theorems''': Brouwer Fixed Point Theorem: any continuous map from a closed disk to itself has a fixed point. Hairy Ball Theorem: there is no non-vanishing continuous tangent vector field on a sphere (you can't comb a hairy ball flat). Jordan Curve Theorem: any simple closed curve in the plane divides it into two regions. Borsuk-Ulam Theorem: any continuous map from Sⁿ to ℝⁿ maps some pair of antipodal points to the same point.
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