Different ways to state the motivation of the definition of the product topology

Solution 1:

In any category (such as the category Set of sets, the category Grp of groups, ...), the product of objects $A_i$, $i\in I$, is an object $P$, together with morphisms (called canonical projections) $\pi_i\colon P\to A_i$ such that for every object $X$ and family of morphism $f_i\colon X\to A_i$, there exists one and only one morphism $h\colon X\to P$ such that $\pi_i\circ h=f_i$ for all $i\in I$.

Spelling this out for the category Top of topological spaces, leads to the well-known concrete construction.

Solution 2:

The product topology is the coarsest topology for which each projection $\pi_i : X \rightarrow X_i$ is continuous.

This makes the product topology the categorical product in the category of topological spaces. That is, for any other space $Y$ with maps $f_i : Y \rightarrow X_i$ we get a unique map $f: Y \rightarrow X$ such that $\pi_i \circ f = f_i$.

Solution 3:

The two biggest reasons I can think of for why the product topology is defined this way:

• Continuous functions Given continuous functions $f_i:Y\to X_i$ for each $i\in I$, there is a unique continuous function $f:Y\to\prod_{i\in I}X_i$ such that $f_i=p_i\circ f$ for all $i\in I$, where $p_i:\prod_{j\in I}X_j\to X_i$ is the projection map. In fact, this uniquely characterizes the product topology.
• Compact sets If each $X_i$ is compact, then Tychnoff's theorem states that $\prod_{i\in I}X_i$ is compact. This is an extremely powerful theorem, and is equivalent to the axiom of choice. This is certainly not true under what some may see as a more "natural" topology on $\prod_{i\in I}X_i$, i.e. the topology generated by $\{\prod_{i\in I}U_i:U_i\text{ open for each }i\in I\}$.

Solution 4:

For the sake of definiteness, I will refer to the name I have seen most commonly used: the product topology is, as others have mentioned, the Initial Topology https://en.wikipedia.org/wiki/Initial_topology with respect to the projections.

A dual concept is that of the Final Topology, which is the finest topology put on the codomain in $f: Y \rightarrow X $ that makes the set continuous.

https://en.wikipedia.org/wiki/Final_topology.

As an example, the quotient topology is the final topology with respect to quotient maps.