What branch of mathematics does coordinate systems belong to?

Question

I'm a high school student and I'm new to mathematics involving coordinates, especially polar coordinate systems and cartesian coordinate systems.

Answer 1

Coordinate systems are not a part of topology, but the (topological) dimension of a space is. Coordinates are a way to transform geometry into algebra and/or analysis.

Answer 2

Via a cartesian coordinate system ( in a Euclidean space)You are able to define a distance, say You have points $x=\begin{pmatrix}x_1\\\cdot\\x_n\end{pmatrix},y=\begin{pmatrix}y_1\\\cdot\\y_n\end{pmatrix}$ given in cartesian coordinates ( in an $n$-dimensional space) their distance is ( for example) given by $$d(x,y)=\sqrt{(x_1-y_1)^2+...+(x_n-y_n)^2}$$ and now You may say a set $M\subset\mathbb{R}^n$ is $\textbf{open}$ if for every point $x\in M$ there is a disk $D_{\varepsilon}(x)$ contained in $M$: $$D_{\varepsilon}(x)=\{y\in\mathbb{R}^n:d(x,y)<\varepsilon\}\subset M.$$ Now by saying which sets are $\textbf{open}$ You define a topology on a set. This is why this notion is called $\textbf{intrinsic}$. This is just one way how cartesian coordinates can be used to define a topology. So there is a link between coordinates and topology, one couldn't say they are part of it, though it is not clear what that is supposed to mean .

Answer 3

@Henno's answer is in many ways perfect; the second sentence in particular distills something I only understood many years after getting my Ph.D., alas. That also means that it may not be completely enlightening for a beginner. At the risk of adding to confusion, let me say a little more:

"Part of" isn't exactly a well-defined notion in mathematics. We use whatever tools are appropriate to get results. In topology, we tend to think of ourselves as "knowing" about $\Bbb R^n$ (we have a lot of useful theorems about the real line, plane, etc.). If we're studying some space $X$ and can find, say, a (nice) correspondence between some subset $V \subset X$ and an open set $U \subset \Bbb R^n$, then we feel we've got an understanding of $V$: it "looks like a bit of Euclidean space." And we might even use coordinates on Euclidean space to say something about points of $V$. So while "coordinate systems" are not part of topology, there are plenty of places in topology books (esp. differential topology books) where you'll see them appear.