  We introduce a procedure to determine the size and shape of normal
neighborhoods in any spacetimes and their dependence on the precision of the
measurements performed by arbitrary observers. As an example, we consider the
Schwarzschild geometry in Riemann and Fermi normal coordinates and determine
the size and shape of normal neighborhoods in the vicinity of the event
horizon. Depending on the observers, normal neighborhoods extend to the event
horizon and even beyond into the black hole interior. It is shown that the
causal structure supported by normal neighborhoods across an event horizon is
consistent with general relativity. In particular, normal neighborhoods
reaching over an event horizon are void of the Schwarzschild coordinate
singularity. In addition, we introduce a new variant of normal coordinates
which we call Fermi normal coordinates around a point, unifying features of
Riemann and Fermi normal coordinates, and analyze their neighborhoods.
