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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?

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  • $\begingroup$ If the $X_k$ are topological spaces, write $\prod_{k \in N} X_k$ for the product topological space. That's the only notation I know. $\endgroup$ – GEdgar Jan 26 '16 at 0:55
  • $\begingroup$ @GEdgar That generally denotes the Cartesian product but not the topology. $\endgroup$ – Henricus V. Jan 26 '16 at 0:56
  • $\begingroup$ By abuse of notation it also denotes the product topology. Similarly in algebra, $\prod_{i\in I}G_i$ for the product of groups, and in basically any category you can mention. It's only ambiguous if some of the underlying sets have different topologies that you need to consider, and then you might have to resort to $\prod_{k\in\Bbb N}(X_k, \tau_k)$ $\endgroup$ – BrianO Jan 26 '16 at 0:57
  • $\begingroup$ I have never seen a "common convention" symbol for the topology itself. In fact, for many other spaces I have not seen a standard symbol for the topology.Even for the "usual" topology on the reals, the word "usual" is the only convention I know of. $\endgroup$ – DanielWainfleet Jan 26 '16 at 2:34
  • $\begingroup$ I'm glad you explained that by "product topology" you mean the Tychonoff product topology, otherwise I would have wondered if you meant the Tychonoff topology or the box topology or something else. I won't try to guess what set-theoretic object you consider to be "the topology" of a topological space; the collection of open sets, the collection of closed sets, the closure operator, the interior operator, the derived set operator, the assignment of neighborhood systems to points, too many possibilities. $\endgroup$ – bof Jan 26 '16 at 6:57
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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.

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