Rigorously defining coordinate systems on physical space

I just started reading about manifolds and learned of coordinate charts. I am trying to mentally piece together my intuitional understanding of coordinates in our physical space (PS) vs the definition of a coordinate chart for a manifold.

Here are my thoughts: Let's say that we have applied a coordinate system to our PS. To me, it seems natural to interpret this as providing a coordinate map from the set of physical points to $\mathbb{R}^3$ with the standard topology. The problem with this interpretation is that it requires us to endow PS with a topology s.t. it is homeomorphic to $\mathbb{R}^3$.

My question: It seems to me that in order to rigorously use coordinate systems in PS we must first define the proper topology on PS. Is this true? If so how does one define this topology and/or how can we prove that there exists a topology on PS s.t. the space is homeomorphic to $\mathbb{R}^3$?

Solution 1:

There are two ways to defining a manifold. In one, you start with a topological space $M$, and then give a collection of chart maps $\{\phi_i:U_i\to\mathbb{R}^n\}$ which are homeomorphisms onto their images. This definition assumes that you have defined a topological structure on $M$ at some point in the past.

The second definition starts with just a set $M$ and a collection of charts that are compatible in the sense that $\phi_i\circ \phi_j^{-1}:\phi_i(U_i\cap U_j)\to \phi_j(U_i \cap U_j)$ is always a homeomorphism. In this case the charts themselves define a natural topology on $M$, so you don't have to worry about having one ahead of time. Then there is a theorem that tells you a manifold of the second type corresponds to a manifold of the first type, so it doesn't matter which definition you are using.

In the case of "physical space", choosing a given coordinate system induces a topology automatically, because we'll say that a subset $V\subseteq U$ of some chart is open exactly when its image $\phi(V)\subseteq \mathbb{R}^n$ is open. The key is that it doesn't matter which coordinate system you use, since most natural coordinatizations of three dimensional space (spherical, cylindrical, Cartesian, etc.) will induce the same topology.