Why are metric spaces non-empty?

Rudin doesn't require that a metric space be non-empty. I agree that there is no good reason for a convention which says otherwise. For example, for those of you who are convinced by such reasons, we want subspaces of metrizable spaces to be metrizable.

What is the name of the vertical bar in $(x^2+1)\vert_{x = 4}$ or $\left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$?

Is there a proof that $\pi \times e$ is irrational?

Continuous function positive at a point is positive in a neighborhood of that point

Prove that $e^x\ge x+1$ for all real $x$ [duplicate]

Primary decomposition of a monomial ideal

Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions

regularization of a divergent integral

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Convergence of tetration sequence.

Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$

How to expand $\tan x$ in Taylor order to $o(x^6)$

Limit involving binomial coefficients: $\lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}}$