Define $f(x,y)$ on the infinite Euclidean plane as: $$f(x,y) = \{1\ {\rm if}\ x\neq y;\ 0\ {\rm if}\ x=y\}$$
I know that f is a metric and any point is a distance of 1 away from any other point not equal to it. I'm trying to visualize what a circle would be. Would there even be a radius since the point of the center of a circle would still be a distance of 1 from any other point on the circle? I'm thinking that a circle would be all possible points in the plane... so it would take up the entire plane. Any help is appreciated. Thanks!
If you define a circle of radius $r$ to be the set of points at distance $r$ from some fixed centre point, then in that metric every circle has radius $1$, and there is one circle of radius $1$ for each point $p$. Specifically, the circle of radius $1$ centred at $p$ consists of all of the other points of the plane. Each point is the centre of one circle, that circle has radius $1$, and every other point of the plane lies on that circle.
As you can see, the notion of circle isn’t very informative here. It’s meaningful, but it doesn’t really tell you much about the space.