Clones of operations of arity \(\omega \) (referred to as \(\omega \) -operations) have been employed by Neumann to represent varieties of infinitary algebras defined by operations of at most arity \(\omega \) . More recently, clone algebras have been introduced to study clones of functions, including \(\omega \) -operations, within the framework of one-sorted universal algebra. Additionally, polymorphisms of arity \(\omega \) , which are \(\omega \) -operations preserving the relations of a given first-order structure, have recently been used to establish model theory results with applications in the field of complexity of CSP problems. In this paper, we undertake a topological and algebraic study of polymorphisms of arity \(\omega \) and their corresponding invariant relations. Given a Boolean ideal X on the set \(A^\omega \) , we endow the set of \(\omega \) -operations on A with a topology, which we refer to as X-topology. Notably, the topology of pointwise convergence can be retrieved as a special case of this approach. Polymorphisms and invariant relations are then defined parametrically with respect to the X-topology. We characterise the X-closed clones of \(\omega \) -operations in terms of \(\textrm{Pol}^\omega \) - \(\textrm{Inv}^\omega \) and present a method to relate \(\textrm{Inv}^\omega \) - \(\textrm{Pol}^\omega \) to the classical (finitary) \(\textrm{Inv}\) - \(\textrm{Pol}\) .
