A book I am reading states that one possible definition of a Bravais Lattice is that the surroundings will look the same from whichever lattice point you observe from.
Consider for example the simple cubic lattice - this definition clearly holds here for any lattice point.
However when considering a BCC lattice it would seem to me that the central lattice points are no longer equivalent to those on the outside - the connections and relative positions to the other lattice points are completely different.
Likewise for an FCC lattice it would seem to me that the face centred lattice points are no longer equivalent to those on the outside in a similar way.
However these structures are called lattices, so there appears to be a contradiction - am I completely wrong or is there something I am missing about the concept of a Bravais lattice?