Consider the following definition:
A profinite group is a topological group that is compact, Hausdorff, and admits a neighborhood basis of $1$ made of normal subgroups.
My question is: it is restrictive to replace neighborhood basis with open neighborhood basis?
In other words, I am trying to see if given a neighborhood basis of $1$ made of normal subgroups is it possible to construct a neighborhood basis of $1$ made of OPEN normal subgroups.
For a generic topological space this is trivial, because given $U_x$ a neighborhood in a neighborhood basis of a point $x$, then by definition there exists an open set $A$ such that $x\in A\subseteq U_x$, and so we can replace $U_x$ with $A$ to obtain an open neighborhood basis. The problem here is that this method doesn't necessarily preserve the normal subgroup structure.