On the topology of non-isolated real singularities

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.