On uniform Structure induced by pseudo metric

Question

If $ U^{'} $ induced by pseudo metric d, then the induced uniform structure on $ X $ induced by the pseudo metric $ d(f\times f) $?

Answer 1

Suppose $f: X \to (Y, \mathcal{U}_d)$ is given for some pseudometric $d$ on $Y$.

So a base of the induced uniformity on $X$ will be all sets of the form

$$(f \times f)^{-1}[U_d(r)] \text{, where } U_d(r)=\{(y,y') \in Y^2: d(y,y') < r\}$$

with $r>0$. Instead, we can define a psueodmetric on $X$ by $d_X(x,x')=d(f(x), f(x'))$ and we have $d_X = d \circ (f \times f)$ as maps and

$$(f \times f)^{-1}[U_d(r)] = U_{d_X}(r)$$ by definition and so we can see this induced uniformity also as the one induced by the pseudometric $d_X$.