To clarify, suppose that the agents are chatbots. Then given a sequence of previous messages M, it outputs a probability distribution over the next message that the agent wants to say. For example, if the task is rock-paper-scissors, it would output a probability distribution with three possible outputs, “rock”, “paper”, and “scissors”, each with 1⁄3 probability.
Under this structure, there is a product topology indexed over the set is sequences of messages.
If you only want to use the structure defined in the post, another topology would be the finest topology that makes your two operations continuous.
To clarify, suppose that the agents are chatbots. Then given a sequence of previous messages M, it outputs a probability distribution over the next message that the agent wants to say. For example, if the task is rock-paper-scissors, it would output a probability distribution with three possible outputs, “rock”, “paper”, and “scissors”, each with 1⁄3 probability.
Under this structure, there is a product topology indexed over the set is sequences of messages.
If you only want to use the structure defined in the post, another topology would be the finest topology that makes your two operations continuous.