In the context of topology, specifically in the realm of product topologies and ideal topologies, the product of ideal topologies may or may not be an ideal topology, depending on the specifics of the situation. Let's break down the concepts involved:

Now, when considering the product of two topological spaces with ideal topologies, the product topology on the Cartesian product of the spaces will not necessarily be an ideal topology. This is because the product topology is defined based on the requirements of continuity for projection maps, and those requirements might not align with the properties of an ideal.

In general, the product of two ideal topologies will likely generate a topology that isn't an ideal topology according to the definition of ideals in topology. However, the specific behavior will depend on the choice of ideal topologies and the space under consideration.

If you have a specific example or scenario in mind, I'd be happy to help analyze it in more detail.

Thank u good comment

What you think abiut product case up to IxI

May or may not be.