General definition of Surfaces in $\mathbb{R}^{n}$

Question

What is a definition of a sphere that contains only topological information? And that of a torus? I can only Think of definitions like "points in $\mathbb{R}^{n}$ that satisfies: $f(x) = 0$" but is there a more General way of defining at Least some Surfaces? Let's take $n=3$ as a reference.

Answer 1

The term "surface" can be defined in subtly different ways depending on the context. Two definitions that seem useful to your purposes are:

A topological $2$-manifold (without boundary) in $\mathbb R^n$ is a set $M\subset \mathbb R^n$ such that, for all $p \in M$, there exists an open set $U$ such that $p\in U$ and $U\cap M$ is homeomorphic to $\mathbb R^2$.

A topological $2$-manifold with boundary in $\mathbb R^n$ is a set $M\subset \mathbb R^n$ such that, for all $p \in M$, there exists an open set $U$ such that $p\in U$ and $U\cap M$ is homeomorphic to $\mathbb R^2$ or to the half-plane $\text{H}^2=\{(x,y)\in \mathbb R^2: y \geq 0\}$.

Sometimes these definitions can be modified and restricted $-$ for example, it's quite common to demand that both the homeomorphism and its inverse be smooth functions, in which case $M$ is called a smooth manifold rather than a topological manifold.

Classifying a particular surface $M$ as a sphere, torus, disc, etc. usually involves treating $M$ as a topological subspace of $\mathbb R^n$ and describing some of the topological invariants of that space in the hope of classifying $M$ up to homeomorphism or up to homotopy equivalence, which is a weaker condition (homeomorphic spaces are always homotopy equivalent, but not vice versa).

Some of these invariants are probably familiar from general topology $-$ e.g. a distinction is often made between compact and non-compact manifolds. Others are less likely to be familiar. The most important ones are probably the homotopy and homology groups associated with $M$, which together characterize it quite extensively. Unfortunately, the definitions are often quite complex and laboured. However, they are closely associated with simpler invariants, such as the genus and the Euler characteristic. The definition of the Euler characteristic, in particular, is both relatively accessible and of considerable historical importance.

One can also talk about $M$ as a topological space on its own without having it inherit a subspace topology from $\mathbb R^n$. This takes more work, but it's possible to topologically characterize surfaces without worrying about what space they're embedded into.

There is quite a lot to say about topological classification of manifolds and entire fields of study dedicated mostly to this topic, so I'm only barely scratching the surface here (excuse the pun). But I hope it is a least a little helpful as a jumping off point.