Different forms of the unit circle

Question

I've been studying topology recently, and I've gotten to the part of the book that deals with quotient spaces. For the most part, it's fairly clear, but one thing that has been confusing me a bit is how the unit circle is represented.

Sometimes $\mathbf S^1$ is denoted as $\{(x,y)\in\mathbb R^2|(x-a)^2+(y-b)^2=1\}$ while other times it's denoted as $\{z\in\mathbb C| \; |z-w|=1\}$. I know these are both representations of the same thing, but I'm not sure whether to consider $\mathbf S^1$ as a subset of $\mathbb R^2$ or as a subset of $\mathbb C$, or if it even matters.

Answer 1

$z$ as a complex has the form $z=x+yi$ for real numbers $x,y$ and $z$ is identified with $(x,y)$ as a coordinate in ${\bf{R}}^{2}$. And the topology in ${\bf{C}}$ is the Euclidean topology for ${\bf{R}}^{2}$. So if $w=(a,b)$, then $|z-w|^{2}=(x-a)^{2}+(y-b)^{2}$, and hence $|z-w|=1$ if and only if $(x-a)^{2}+(y-b)^{2}=1$.

Answer 2

These are different only on the level of algebra.
From geometric point of view, these are exactly the same (with $w=a+bi$ correspondence), using the natural homeorphism $\Bbb R^2\cong\Bbb C,\ \ (x,y)\mapsto x+yi$.

Answer 3

$$\{(x,y)\in\mathbb R^2|(x-a)^2+(y-b)^2=1\}$$ is the same set as$$ \{z\in\mathbb C| \; |z-w|=1\}$$ with $w=a+bi.$

For the subject of metric topology it is probably easier to talk about d(z,w) = |z-w| than using the square root notation.

Answer 4

$S^1= \{ (x,y) \in R^2 : x^2 + y^2 = 1 \}.$
All those other things are just homeomorphs, isomorphs even.
Whether it is a subset of the real plain or the complex plain depends upon the context in use.