Homotopy properties of algebraic sets

In this paper there are announced several homotopy theorems on algebraic subsets of complex projective spaces. Some of the theorems generalize and refine results of the preceding note of the author (RZhMat, 1978, 5A531). It is also asserted that any $n$-dimensional complex affine algebraic set has the homotopy type of a finite $n$-dimensional cellular space. A generalization is given, in two versions, of a theorem of Bart–Larsen (RZhMat, 1973, 7A587).