Semicontinuity of Betti numbers

A consequence of Thom Isotopy Lemma is that the set of solutions of a regular smooth equation is stable under C^1-small perturbations (it remains isotopic to the original one), but what happens if the perturbation is just C^0-small? In this case, the topology of the set of solution may change, but it turns out that the Homology groups cannot "decrease". In this talk I will present such result and some related examples and applications. This theorem is useful in those contexts where the price to pay to approximate something in C^1 is higher than in C^0. For instance in the search for quantitative bounds (here the price can be the degree of an algebraic approximation) or in combination with Eliashberg's and Mishachev's holonomic approximation Theorem (which is C^0 at most).