The product topology on a Cartesian product of $n$ Topological spaces ($X_i, \tau_i$, $i \in I$, where $I$ is some index set) is defined to be the union of all sets of the form $O_1 \times O_2 \times ... \times O_n$ where $O_i \subset X_i$ is $\tau_i$-open. Where we are assuming here $I$ is finite. This definition is not correct when $I$ is infinite, and the definition using cylinder sets below must be used. Note that the definitions are different because the basis is constructed from finite intersections of the open cylinders. However, some elements corresponding to infinite Cartesian products of the form $\times_{i\in I} O_i$ can't be realized from finite intersections of open cylinders which all have the form $(\times_{i = J} O_i) \times ( \times_{i\in I \setminus J} X_i)$, where $J$ is a finite subset of $I$. This comes about for example, in infinite Sequence spaces.

Motivation – definition (vid)

It can also be constructed using Filter subbases and Filter bases (that generate the open sets of the topology)

Note the elements $U(j, O)$ forming the subbase are part of the final topology. They have the form $O_1 \times O_2 \times ... \times O_n$ described above if we rememeber that the full set $X_i$ is always open.

The sets forming the subbase are known as open cylinders, while those forming the basis are known as Cylinder sets.

Another equivalent way of defining the product topology is as the 'smallest' topology such that the projection functions $\pi_j:\times_i X_i \rightarrow X_j$, $f \mapsto f(j)$ are Continuous functions.

A smaller subbase (in the case of a product of finite sets) is given by the sets defined here. They are like open cylinders but where we require $f(j)$ to belong to the basis of $X_i$ instead of to its topology (Warning, nonstandard terminology: He calls them cylinder sets, but that doesn't agree with the standard terminology, used here. We call them Basic open cylinder).

video

Proposition to characterize Continuous functions when the codomain is a space with Product topology

Product topology is Metrizable on $\mathbb{R}^\omega$, where $\omega$ is a countable index set

Product topology is not Metrizable on $\mathbb{R}^J$, when $J$ is Uncountable