- There are connected surfaces (sphere) and non-connected surfaces (hyperboloid of two sheets, with two connected components).
- There are compact surfaces (sphere) and non-compact surfaces (plane).
- There are closed surfaces (sphere) and non-closed surfaces (a hemisphere).
- The boundary of the sphere S2 is the very sphere S2.
- Every point of a surface has a neighborhood homeomorphic to R2. In fact, the coordinate open V of p∈S is homeomorphic to an open set U⊂R2 via the parametrization X:U→V. Since U is an open set, there exists a ball Br(q) around q=X−1(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={p∈S2: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.
- If ϕ:R3→R3 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:S→ϕ(S).
- An open set of a surface is a surface (proved).
- 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.
A blog of the course Curvas y superficies of the 2º-B classroom of the University of Granada, Spain
Tuesday, 14 March 2017
Surfaces and topology
The surfaces that we are introduced present a variety of possibilities on its topology.
Labels:
connectedness,
interior,
surface,
topology
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