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Tuesday, 14 March 2017

Surfaces and topology

The surfaces that we are introduced present a variety of possibilities on its topology.

  1. There are connected surfaces (sphere) and non-connected surfaces (hyperboloid of two sheets, with two connected components).
  2. There are compact surfaces (sphere) and non-compact surfaces (plane).
  3. There are closed surfaces (sphere) and non-closed surfaces (a hemisphere).
  4. The boundary of the sphere S2  is the very sphere S2.
  5. Every point of a surface has a neighborhood homeomorphic to R2. In fact, the coordinate open V of pS is homeomorphic to an open set UR2 via the parametrization X:UV. Since U is an open set, there exists a ball Br(q) around q=X1(p) with Br(q)U. Then then restriction X|Br(q):Br(q)X(Br(q)) is a homeomorphism, being X(Br(q)) an open set of V, so, of S. This means that X(Br(q)) is an open set around p homeomorphic to R2. As a consequence, we conclude:
    • The interior of a surface is empty.
    • The surfaces 'have not boundary point', I mean, for example, the closed hemisphere T={pS2:z(p)0} is not a  surface because the above property fails at the points with z(p)=0.
    • A point is not a surface.
    • A surface has a non-countable set of points.
  6. If ϕ:R3R3 is a diffeomorphism and S is a surface, then ϕ(S) is a surface which is a homeomorphic to S thanks to the restriction ϕ|S:Sϕ(S).
  7. An open set of a surface is a surface (proved).
  8. Some closed sets of a surface are surfaces; other not. For example, if S is the union of two disjoint spheres, then each sphere is closed and it is a surface. On the other hand, the closed hemisphere is closed in S2 and it is not a closed set.

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