We introduce topological rewriting systems as a generalisation of abstract rewriting systems, where we replace the set of terms by a topological space. Abstract rewriting systems correspond to topological rewriting systems for the discrete topology. We introduce the topological confluence property as an approximation of the confluence property, and use it to characterise standard bases in terms of rewriting theory. Using a representation of linear topological rewriting systems with continuous reduction operators, we also interpret the topological confluence property in terms of lattice operations. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras, from which we deduce a duality for proving that an algebra is syntactic or not.