Suppose there are $n$ metric spaces $(X_i,d_i)$, and $X$ being the product space with the product metric $\rho$.
In proving that the projection function $\pi_i(x_1,...,x_n) = x_i$ is continuous, the proof follows something like this (not rigorous since it's only serving to motivate the question):
Fix $\varepsilon > 0$. Then we get the existence of $\delta := \varepsilon *2^{-i}$ such that
$\rho(x,y)$ = $\sum_{j=1}^{n} 2^{-j} min(1, d_j(x_j,y_j))$ < $\delta$
Implies that, for a specific $i$,
$2^{-i}d_i(x_i,y_i) \leq \sum_{j=1}^{n} 2^{-j} min\{1, d_j(x_j,y_j)\} < \delta$ (1)
So $d_i(x_i,y_i) < \varepsilon$ and the function is continuous.
In this proof, it actually follows that we can, without lose of generality, consider $min\{1,d_i(x_i,y_i)\}$ as $d_i(x_i,y_i)$, in (1). I would like to know why that is the case.
Thanks
Let p = $\pi_i:X_1×..×X_n$ -> $X_i, (x_1,.. x_n)$ -> $x_i.$
To show p is continuous, let U be any open subset of $X_i$
and show $p^{-1}(U) is open. This will also show
projections of infinite products are continuous.
In fact without much ado the projections can be shown to be open.