I'm reading through Schaum's General Topology and I came across the section on equivalent metrics. So, the author introduces the idea that metrics $d_1$ and $d_2$ are equivalent when the $d-$open spheres of $d_1$ and $d_2$ induce the same topology on a space $X$. Then, as an example, the author goes on to show that the three metrics below all serve as a base for the natural topology on $\mathbb{R}^2$.
So, I can clearly understand how we can use these shapes as a base for the natural topology on $\mathbb{R}^2$. What I don't understand is how $S_{d_1}(p,\delta)$ and $S_{d_2}(p,\delta)$ are open "spheres". Geometrically speaking $S_{d_1}(p,\delta)$ and $S_{d_2}(p,\delta)$ are squares. Since they are squares, they are not "spheres" in the sense that they are sets centered at some point $p$ which contain all points within $\delta$ of $p$. Like I said, squares are perfectly fine as a base for the natural topology on $\mathbb{R}^2$ but can we really call $S_{d_1}(p,\delta)$ and $S_{d_2}(p,\delta)$ open spheres? In a square centered at $p$, won't some points be farther from $p$ than others, thus violating our traditional notion of open sphere?
Best Answer
Spheres in metrics do not mean a physical sphere, i.e. a round circle or something like that.
In fact, usually, a sphere is an open neighborhood of a certain point that contains all points which have a distance to the center point lower than $\delta$. The key observation is that the "distance" is just a function (which has certain properties s.t. it's called "metric"), but it doesn't have to be the Euclidean distance.
You are imaging a circle - that is using the Euclidean distance as your metric. However, the square comes by using the metric $d((x_1,y_1),(x_2,y_2)) = \max\{|y_2-y_1|,|x_2-x_1|\}$. The shape of the set of all points with $d(x,y) < 1$ is indeed a square, which is, nevertheless, called the "sphere" around a point. That may seem confusing at first glance, but working with topology, spheres can have all kind of shapes, which could be non-circular as well.
In general open spheres, topologically speaking, are subsets in a metric space $(R,d)$ of the form $S_d(p,\delta) = \{q \in R| d(p,q)<\delta\}$, whereas closed spheres are $S_d(p,\delta) = \{q \in R| d(p,q)\leq\delta\}$.
In your case we have e.g. $R=\mathbb{R}^2$.