You could identify 0 with some other point to get a topology where {0} is not a Borel set anymore. A somewhat more useful example would be to identify x with x

You could identify 0 with some other point to get a topology where {0} is not a Borel set anymore. A somewhat more useful example would be to identify x with x+1 for all x (or equivalently use the topology on R induced by the map t -> e^2πit from R to S^1), which results in a topology where only periodic functions are continuous. For a complete measure you get the additional requirement that 0 isn't allowed to be contained in any null-set.