In profinite group $G$ with inverse system $(G_\alpha,\phi_{\alpha,\beta} )%$, how the kernel of projection map $G\rightarrow G_\alpha$becomes the basis of neighborhood for $1$ in $G$?
I couldn't understand this sentence in p.21 of the book 'Galois groups and Fundamental groups, Tamas Szamuely'.
Recall the construction of the topology on a profinite group (the projective limit $P$ of a directed set of finite groups $G_\alpha$): all the groups $G_\alpha$ are given the discrete topology, so that in particular $\{1_{G_\alpha}\}$ is open. The projective limit then has the subspace topology inside $\prod_\alpha G_\alpha$.
Let us set some definitions straight:
An open set in the product topology of $\prod_\alpha G_\alpha$ is of the form $\prod_\alpha U_\alpha$, where $U_\alpha\subset G_\alpha$ is open and $U_\alpha=G_\alpha$ for all but finitely many $\alpha$. In other words, it is a finite intersection of sets of the form $\pi_\alpha^{-1}(U_\alpha)$ for $U_\alpha\subset G_\alpha$ open.
A neighbourhood basis for $1$ in the projective limit $P$ is a set of open neighbourhoods $\mathscr{N}$ such that for any open subset $U\subset P$ that contains $1$, there exists $ N\in \mathscr{N}$ that is contained in $U$.
Can you now see that the open sets $\pi_\alpha^{-1}(\{1_{G_\alpha}\})=\operatorname{Ker} \pi_\alpha$ form a neighbourhood basis for $1$ in the profinite group?