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. Using a representation of linear topological rewriting systems with continuous reduction operators, we show that the topological confluence property is characterised by lattice operations. Using this characterisation, we show that standard bases induce topologically confluent rewriting systems on formal power series. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras. In particular, we use duality for proving that an algebra is syntactic or not.