  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 propose a method to 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
$Pol^\omega$-$Inv^\omega$ and present a method to relate
$Inv^\omega$-$Pol^\omega$ to the classical (finitary) $Inv$-$Pol$.
