Let $((X_k,\tau_k))_{k \in N}$ be topological spaces. The product topology $\tau$ on $X = \prod_{k \in N} X_k$ is the coarsest topology that makes all projections $\pi_k:X \to X_k$ continuous.
Is there a notation that I can use for $\tau$ such as $\tau = \bigotimes_{k \in N} \tau_k$? Or is there no convention for product topology?
Best Answer
No one ever uses a symbol for the product topology. When anyone in almost any circumstance works with the set $\prod\limits X_k$ and assumes it is a topological space, they are tacitly assuming it is taken in the product topology, unless they say otherwise.
There is a notable exception in algebraic geometry, where if $X$ and $Y$ are varieties over an algebraically closed field (topological spaces with some extra structure), then the cartesian product $X \times Y$ can be given the structure of a variety, and the projection maps $X \times Y \rightarrow X, Y$ are continuous. In making $X \times Y$ into a variety (called the product variety), it is given a topology, but this topology almost never coincides with the product topology on $X \times Y$. However, this topology contains the product topology.