  Khimshiashvili proved a topological degree formula for the Eu-ler
characteristic of the Milnor fibres of a real function-germ with an isolated
singularity. We give two generalizations of this result for non-isolated
singularities. As corollaries we obtain an algebraic formula for the Euler
characteristic of the fibres of a real weighted-homogeneous polynomial and a
real version of the L{\^e}-Iomdine formula. We have also included some results
of the same flavor on the local topology of locally closed definable sets.
