This paper deals with tropical algebraic sets with the structure of an idempotent semigroup and their topology. After general results about the topology of idempotent semigroups, the focus is on intersections of tropical hypersurfaces which are called tropical algebraic sets. If a tropical algebraic set is a subsemigroup of \(\mathbb{R}^n\) with the tropical addition, it is called an additive tropical set. The support of a weighted balanced polyhedral complex is called a tropical set-variety. A tropical polynomial is called simple if its monomials are univariate or constant. The authors prove that any tropical algebraic set defined by simple polynomials is additive and conjecture the reverse statement. They prove their conjecture for hypersurfaces, tropical algebraic sets defined by binomials (i.e. usual affine subspaces), plane curves and spatial curves. Finally, they show that the skeletons of an additive tropical set-variety (where the polyhedral structure is naturally defined) are additive, thus their connected componenents are contractible using the general topological results.