On every metric space $(T, d)$ metric $d:T^2 \to \mathbb{R}$ is continuous function?

Question

On every metric space $(T, d)$ metric $d:T^2 \to \mathbb{R}$ is continuous function? How to prove it?

Answer 1

The space is metric, so $T\times T$ is Hausdorff and first countable. This means that a map $f$ is continuous if and only if $\{a_n\}_n$ it is convergent to some $a$, then $f(a_n)\to f(a)$.

Now we consider the function $f=d$ and a sequence $(a_n,b_n)$ convergent to some $(a,b)$. We fix $\epsilon>0$ Then there exists some $N>0$ for which

$|d(a_n, b_n)-d(a,b)|<\epsilon$

for each $n\geq N$.

Then

$|f(a_n,b_n)-f(a,b)|<\epsilon$

for each $n\geq N$.

Thus $f(a_n,b_n)\to f(a,b)$ and this means $f=d$ is continuos on $T\times T$.