Definition talk:Product Space (Topology)/Two Factor Spaces

Tychonoff Topology
Let $\mathcal{T}$ be the Tychonoff topology on $X$. Since $\mathcal{T}$ is the topology generated by $\mathcal S = \left\{{\operatorname {pr}_i^{-1} \left({U}\right) : i \in I, U \in \vartheta_i}\right\}$ isn't it also the topology that renders all projections continuous (linear) functionals? Therefore, $\mathcal{T}$ is the product topology on $X$. What one needs to prove is that the functions:
 * $\pi_1(x)=x_1$

and
 * $\pi_2(x)=x_2$

where $x\in X$ is $x=(x_1,x_2)$ with $x_1\in X_1$ and $x_2\in X_2$,

are continuous with respect to the described topology with basis $\mathcal{B}$. However, I don't like very much this definition because firstly it refers only to cartesian products between two topological spaces (while you can have Cartesian products made up of arbitrarily many such spaces) and because most textbooks provide the definition that the product topology is the topology that renders all projections continuous (i.e. exactly this definition). For example S. Axler and K.A. Ribet, "A Taste of Topoology", Springer Editions, Berlin 2005, ISBN: 0-387-25790-X. I suggest that we modify the definition of Product topology and have just a link to the Tychonoff topology.
 * While I agree with most of what you say, some remarks are in order. First of all, the projections aren't functionals in the functional analytic sense of the word (which is the only one known to me), they are operators. Linearity subsumes an additive structure, which isn't given. Lastly, please be aware that the definition of an arbitrary Cartesian product needs the Axiom of Choice to render it nonempty. --Lord_Farin 12:15, 30 November 2011 (CST)

Refactor
The refactor comment suggests creating theorems for the statements:


 * The product topology $\tau$ is the same as the box topology for $S_1 \times S_2$.


 * It is also the same as the Tychonoff topology for $S_1 \times S_2$, which follows from Box Topology on Finite Product Space is Tychonoff Topology.

which seemed to me to be unnecessary since Box Topology on Finite Product Space is Tychonoff Topology already does this. Instead it seemed more appropriate to cretae the page with two definitions and state that the equivalence of the definitions was given by Box Topology on Finite Product Space is Tychonoff Topology.

This is what I have done on this page Leigh.Samphier/Sandbox/Definition:Product Space (Topology)

If this is a suitable alternative, let me know and I'll put these pages in place. --Leigh.Samphier (talk) 04:05, 17 December 2019 (EST)