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

## Product Topology

Let $\TT$ be the product topology on $X$. Since $\TT$ is the topology generated by $\SS = \set {\map {\pr_i^{-1} } U: i \in I, U \in \vartheta_i}$ isn't it also the topology that renders all projections continuous (linear) functionals? Therefore, $\TT$ 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 $\BB$. 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 Topology", 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 product 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 product topology for $S_1 \times S_2$, which follows from Box Topology on Finite Product Space is Product Topology.

which seemed to me to be unnecessary since Box Topology on Finite Product Space is Product 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 Product Topology.

This is what I have done on this page 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 Product topology more specific, intimately connected. At present, they are unjustly disjoint.
Maybe the most pure solution would be to transclude both Product topology 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? — 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:Product 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 Product Topology be the topology only and Product Space be the space with the Product 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? — 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:Product Topology and Definition:Box Topology.

My proposed new versions are:

Definition:Product Space (Topology)
Definition:Product Topology
Definition:Box Topology

respectively.

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

The pages Definition:Product Topology and Definition:Box Topology have notes added to walk the reader from the definition of the Product topology as an initial topology to more familiar definitions in terms of basis of products of open sets. I have also clarified what the Product 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 Natural Basis of Product 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. — Lord_Farin (talk) 04:12, 30 December 2019 (EST)
Ok, so I have a few remarks:
• The first two sentences under "Note" say not much more than a link under Also see would.
• I would relegate the statement of the exact nature of natural basis and subbasis to the respective subpages and/or subsume it in the Also see section as well.
• The relation to the box topology is nice. I am however not sure how to best structure the pages so that the reference to it from both Product and box topology are natural (subpages are not very suitable here I feel). The links to the "may not" results can then also reside on this subpage. @Prime.mover, do you have an idea how to structure this? Surely there are precedents.
Maybe we could make an overview page titled "Relation between Product and Box Topology" or some such?
Split the "Also see" into subsections. (There's a precedent -- Barto started this convention when he was focusing on raising the importance of this section). Put one section as "Box Topology" and in it list (and briefly describe -- the usual one-liner) what the various pages contain. (We may even want to put that "also see" into its own transcluded subpage -- again, there are precedents for this, can't immediately place one, but it's something we've done.)
As it stands, the section in question works well enough as it is. That is, taking Definition:Product Topology as an example: "Also see" starts at "Note" (which is renamed "Also see") and ends where the existing "Also see" is, which only contains the existing links anyway.
See how it looks when it's done. We can tinker with it then.
BEWARE SPELLO: "Box Topology may not be Coursest Topology such that Projections are Continuous" -- that is "Coarsest". --prime mover (talk) 17:09, 30 December 2019 (EST)
• Please assess to the best of your ability the impact of these changes on any page referring to any impacted page before effecting the change. You don't have to list it exhaustively for me, but I've always found it convenient to make a list of pages to be updated/restated.
Overall it looks nice and structured. Thanks for your efforts! — Lord_Farin (talk) 09:35, 30 December 2019 (EST)
I'll take all that on board and try and create a common page that can be transcluded by both the Product Topology page and the Box Topology page. I did try this but it wasn't working for me at the time.
Regarding impacted pages I will certainly do this. The biggest change is on the page Definition:Product Space (Topology) where it is now defined in terms of the Product Topology (like the general case) rather than the basis of cartesian products of open sets of the factor spaces. I do immediately invoke the theorem Natural Basis of Product Topology of Finite Product to define a basis for the Product topology. This is so that any link to the page that assumes that the product topology is defined by the basis of cartesian products of open sets of the factor spaces is still free to do so. Is this Ok? Or does an invocation to Natural Basis of Product Topology of Finite Product need to be made by pages linking to the page and assuming the basis? --Leigh.Samphier (talk) 16:42, 30 December 2019 (EST)
Thanks for taking this task on. As I've always said, it's easy to just write up pages containing mathematics, but it's surprisingly difficult to structure it. But then it takes a huge effort to make a task look effortless. --prime mover (talk) 17:12, 30 December 2019 (EST)
I think the latest versions are closer to what has been suggested. The pages that I have altered or added are:
--Leigh.Samphier (talk) 00:50, 31 December 2019 (EST)

I like the structure of the page "Relation between Product and Box Topology". I've put a mergeto note on the Universal Property page linked there as it is just the statement of the categorical product and should be subsumed in there (a redirect can remain).

I am however not so sure it should be the "Also see" section (NB: note the capitalisation of "Also see") because this can create unexpected effects (e.g. Box topology-specific references under the Product page). Instead I would prominently link to the "Relation" page in the Also see section of Product and Box topology.

Besides that there might be minor stylistic points but as Prime.mover suggested they can be dealt with after the shift has taken place. Good job! — Lord_Farin (talk) 07:43, 31 December 2019 (EST)

Ok, I've now created Universal Property of Product of Topological Spaces and Definition:Product Space of Topological Spaces as redirects. I've dropped the tranclusion of the page Relation between Product and Box Topology from both Definition:Box Topology and Definition:Product Topology and replaced with a prominent link.
The pages to be moved and added are now:

The following pages list the further changes that I believe need to be made to align with the changes above.
• "Pages Impacted: Box Topology"
• "Pages Impacted: Product Topology"
• "Pages Impacted: Product Space"

I've had enough for the first day of 2020. --Leigh.Samphier (talk) 01:31, 1 January 2020 (EST)

I have now completed the changes. --Leigh.Samphier (talk) 22:37, 4 January 2020 (EST)

Superb job. --prime mover (talk) 01:20, 5 January 2020 (EST)

## Further work

I have embarked on the task of merging Definition:Product Topology and Definition:Product Space (Topology) into one actual entity.

This is so as to make it completely clear that they are one and the same thing conceptually: the one is the topological space induced by the other.

Thus the main definition will be as the coarsest topology on the cartesian product of an arbitrary family of sets.

The 2-factor instance of this will be entered as a subpage. This is as opposed to on the Definition:Product Space (Topology) page where the general form is as a subpage of the 2-factor instance.

In this way it will be easier to emphasise the definition of the natural basis and natural sub-basis, which may also be on the way to being refactored.

This may take a day or two, and I have a day job and a social outlet taking up most of my time tomorrow, so don't expect this to be complete much before the weekend. --prime mover (talk) 22:39, 8 December 2020 (UTC)

I think what you are proposing is a good idea. It appears to be more fashionable to refer to the coarsest topology on the cartesian product as the product topology and the resulting space as the product space. This aligns well with this space being the categorical product, and hence why product topology is more fashionable. It would be good to retain a historical note page for the Tychonoff Topology that recogonised this as the same thing as the product topology and transcluded the definition product topology. More people will be searching for the product topology these days than the Tychonoff topology, but at the same time you don't want to miss any search for Tychonoff --Leigh.Samphier (talk) 00:00, 9 December 2020 (UTC)
Major renaming under way. Some of the above will become inaccurate as I change the names of the pages themselves with a view to avoiding redlinks as and when some of the confusing renames are cleaned up. The aim is that much of the above discussion will be removed and the resulting work summarised. --prime mover (talk) 07:26, 9 December 2020 (UTC)
Right, I think I've hammered it into the shape I envisaged.
Having done so, I'm not sure now whether it's best this way -- we need to evaluate whether it's best:
a) making the general case the main page, with a subpage of "2 factor space"
b) maing the main page the 2-factor space and the subpage the "General case" (which is the way it was before).
Thoughts on which way round is best? The main disadvantage of the first case is that the page names becomes unwieldy -- although most of the results should relate to the general case so this may not be all that much of a problem.
I also note there may also be a need to document the general product space for a general finite cartesian product, as this is a significant case which is explored in its own right and merits its own page.
Apologies for having taken so long to get round to it, I have finally got my head round understanding the nature of product spaces, it was like a light going on. --prime mover (talk) 08:59, 10 December 2020 (UTC)