The transition function in a Markov decision process

Question

A Markov decision process is typically described as a tuple $\langle A,U,T,R \rangle $ where

• $A$ is the state space
• $U$ is the action space
• $T: A \times U \times A \mapsto [0,\infty) $ is the state transition probability function
• $R:A \times U \times A \mapsto \mathbb{R}$ is the reward function

What does this $A \times U \times A$ actually mean in terms of the MDP? It is written in all the papers, but never explained. Does it mean that all the states $a \in A$ are multiplied with all the action $u \in U$? Or something completely different?