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 ${\mathbb S}^2$  is the very sphere ${\mathbb S}^2$.
5. Every point of a surface has a neighborhood homeomorphic to ${\mathbb R}^2$. In fact, the coordinate open $V$ of $p\in S$ is homeomorphic to an open set $U\subset{\mathbb R}^2$ via the parametrization $X:U\rightarrow V$. Since $U$ is an open set, there exists a ball $B_r(q)$ around $q=X^{-1}(p)$ with $B_r(q)\subset U$. Then then restriction $$X_{| B_r(q)}:B_r(q)\rightarrow X(B_r(q))$$ is a homeomorphism, being $X(B_r(q))$ an open set of $V$, so, of $S$. This means that $X(B_r(q))$ is an open set around $p$ homeomorphic to ${\mathbb R}^2$. 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 $T=\{p\in{\mathbb S}^2: z(p)\geq 0\}$ 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 $\phi:{\mathbb R}^3\rightarrow{\mathbb R}^3$ is a diffeomorphism and $S$ is a surface, then $\phi(S)$ is a surface which is a homeomorphic to $S$ thanks to the restriction $\phi_{|S}:S\rightarrow \phi(S)$.
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 ${\mathbb S}^2$ and it is not a closed set.