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)


 * I would say that there do not need to be two definitions. Instead I would make the connection between product space and Tychonoff product more specific, intimately connected. At present, they are unjustly disjoint.


 * Maybe the most pure solution would be to transclude both Tychonoff product and box topology onto a general, almost disambiguation-style page that highlights the similarities and differences.


 * By appropriately distinguishing the finite and infinite cases, we could provide proper guidance of intuition while still remaining precise in the infinite case. Some examples would really finish the deal.


 * What do you say? &mdash; Lord_Farin (talk) 13:26, 17 December 2019 (EST)


 * A worthy aim but challenging, and needs someone who knows their way around. Well volunteered, Leigh! :-) --prime mover (talk) 14:39, 17 December 2019 (EST)


 * That has me thinking. I'm not convinced that disambiguation is required, so I may not be the person to refactor this.


 * In my opinion when someone searches for Product Space or Product Topology they want the initial topology with respect to the projections because this is the categorical product whether they know this or not. They are very unlikely to want the box topology.


 * Someone encountering the Product Space for the first time may find the initial topology with respect to the projections definition daunting and the box definition on a finite set of topologies would be more easily understood and all that is required. But irrespective of the definition it is the initial topology that is being looked for. The box topology is only of interest as an aside to the general definition of the product topology on an infinite cartesian product to emphasise and contrast with the initial topology definition.


 * If I had a blank slate, I would create two pages:
 * (1) Product Space/Product Topology
 * (2) Box topology
 * there would be no Tychonoff Topology page, this would just be an 'Also known as' on the Product Space page.


 * The Product Space page would have a general definition and a finite definition. The general definition would have two definitions:
 * (1) The initial topology with respect to the projections
 * (2) The set of cartesian products of open sets where only finitely many open sets are not the complete space.


 * For the finite definition, the box definition then becomes a special case of the second definition of the general definition.


 * Both pages could then have a note that states that the two definitions on the finite Cartesian product of spaces define the same topology.


 * But I don't have a blank slate. The current state is that there are two definitions and 3 pages. And I'm not sure what is trying to be achieved by that. The pages Definition:Tychonoff Topology and Definition:Product Space (Topology) are duplicates as they both define the topology and the space with the topology. So either they should be merged or keep the two pages but have Tychonoff Topology be the topology only and Product Space be the space with the Tychonoff Topology only.


 * Lots to be thought about. --Leigh.Samphier (talk) 05:49, 18 December 2019 (EST)


 * All things considered, I agree with you that there should not be 3 pages. The Tychonoff topology should stay because we prefer historical naming, product space makes sense because it is the product. Things can be kept simple because we don't talk about the "box space".


 * To keep things accessible, I would keep the distinction between the binary and general product as separate subpages.


 * Do we agree enough for you to draft a suggestion based on our discussion? &mdash; Lord_Farin (talk) 13:34, 18 December 2019 (EST)


 * I think I have enough to have another attempt. --Leigh.Samphier (talk) 18:24, 18 December 2019 (EST)

Refactor - Take 2
I have made a second attempt to rework the pages Definition:Product Space (Topology), Definition:Tychonoff Topology and Definition:Box Topology.

My proposed new versions are:
 * Leigh.Samphier/Sandbox/Definition:Product Space (Topology)
 * Leigh.Samphier/Sandbox/Definition:Tychonoff Topology
 * Leigh.Samphier/Sandbox/Definition:Box Topology

respectively.

On Leigh.Samphier/Sandbox/Definition:Product Space (Topology) I have defined the product space as the Cartesian product with the Tychonoff topology. No mention is made of the box topology. The theorem Leigh.Samphier/Sandbox/Natural Basis of Tychonoff Topology of Finite Product expresses the basis for the topology on the product space in more familiar terms.

The pages Leigh.Samphier/Sandbox/Definition:Tychonoff Topology and Leigh.Samphier/Sandbox/Definition:Box Topology have notes added to walk the reader from the definition of the Tychonoff topology as an initial topology to more familiar definitions in terms of basis of products of open sets. I have also clarified what the Tychonoff topology gives in terms of the Categorical Product of topological spaces and that the box topology does not give us.

With the proposed changes above, the page Product Topology is Topology probably needs to be reworked. I would suggest that the page be renamed to something like Product of Open Sets is Basis for Topology on Cartesian Product and the theorem reworded as such. A second proof could be added to state that it is a direct consequence of Leigh.Samphier/Sandbox/Natural Basis of Tychonoff Topology of Finite Product and retain the proof from W A Sutherland.

The page Projection from Product Topology is Continuous would need to be reworked to simply state that this follows by definition as the proof for the general case does: Projection from Product Topology is Continuous/General Result. Although if these pages are merged with Projection from Product Topology is Open into Projection from Product Topology is Open and Continuous then the theorems would have a little more substance.

Otherwise, I haven't seen anything else that is significantly impacted by the proposed changes. But I'm still looking.

But its time for some feedback before going to much further. --Leigh.Samphier (talk) 02:18, 29 December 2019 (EST)


 * I'm going to leave that to someone else, as my own headspace is getting too small. --prime mover (talk) 06:38, 29 December 2019 (EST)


 * I hope to attend to it later today. &mdash; Lord_Farin (talk) 04:12, 30 December 2019 (EST)