Talk:Product topology

From formulasearchengine
Jump to navigation Jump to search

Template:Maths rating

the second definition, defining open sets as generated by counterimages of open sets in *each* Xi seems wrong, or maybe I have misunderstood.83.211.54.72 08:55, 18 October 2005 (UTC)

Yes, I misunderstood, it's ok, sorry 83.211.54.72 08:57, 18 October 2005 (UTC)


A product of (non-empty) locally compact spaces will be locally compact exactly in the case all but finitely many are in fact compact

---

Using the adele ring as an illustration of this - it is a so-called restricted product, union of such locally compact subspaces. Does restricted product have wider status?

Charles Matthews 10:14, 8 Nov 2003 (UTC)

Separation axioms

This article currently has a section of separation axioms; It states "every product of Ti spaces is Ti", etc. Clear, maybe for a finite product, or a countable product; are these true even when the index set is uncountable? If so, it would be nice to strengthen the wording of this section to indicate that the claims are true for any product, even uncountable products. linas 22:03, 20 November 2005 (UTC)

We should get some short proofs for these, then we wouldn't be in doubt. --MarSch 12:53, 12 March 2007 (UTC)

unit for topological product

Is the point space a unit of taking the topological product?

Like . This would mean that each space is homeomorphic to a product space. Thus each fiber bundle would be homeomorphic to a product, which would mean wording like "non-trivial product" would have to be used to characterize them.--MarSch 12:52, 12 March 2007 (UTC)

Minor improbements required

My expertise in maths is barely enough to compreend the text, but not to fix it so I have to ask someone else.

  • Pullback must be disambiguated in the intro
  • "box topology is too fine" what would this mean "too" for what? Also, box topology explains this better and in more details. Please sync them.
    • (Oh, I just have noticed this issue is repeated below, and adjacet to in is another unclear phrase: "forms a basis of .... box topology". Please clarify / disambiguate the term "basis" here.
  • I would also avoid the usage of words in quotation marks, unless you write an articles which explain the expectations of a topologist from the words "correct" and "natural". For a person like me I fail to fully comprehend why box topology is less "correct".
  • An example when box topolofy is different from product topology would be handy.

I have more problems with this text, but I am no longer sure whether it it the text or my rusty brain. Thank you. `'Míkka 15:17, 28 August 2007 (UTC)

Responding to each of your points:
  • So do it; it's not hard.
  • The entire sentence reads: However, the product topology is "correct" in that it makes the product space a pullback of its factors, whereas the box topology is too fine....Clearly, it is too fine to make the product space a pullback. However, neither this article nor box topology explain why this is an impediment (it means that trying to define a function into the product space by specifying its coordinates won't always work: even if the coordinate functions are continuous, the product function won't be). That could bear some explanation in either, or both, articles.
  • The two words in the intro which are in quotes were put there to emphasize a comparison. The sentence with "correct", quoted above, puts it in quotes to emphasize that the word is subjective and context-dependent, and the context in question is that of a pullback. "Natural", later in the sentence, was quoted to contrast it with the use of the same word, unquoted, earlier in the introduction. Essentially: you want an explanation, but you haven't noticed that the sentence itself is the explanation.
  • The two differ for any infinite product. This is mentioned in both articles, and although neither one gives an example specifically devoted to this point, you can peruse the one in box topology about the Hilbert cube for more enlightenment.
In general: yeah, both articles suck relatively badly. They are thrown-together and overly technical. They don't have good examples and they emphasize the wrong points at the wrong times, presume inconsistent levels of preparation, and don't even feel complete. I still stand by what I wrote, however. Ryan Reich 17:24, 29 August 2007 (UTC)

say what?

The definition now reads

Let I be a (possibly infinite) index set and suppose Xi is a topological space for every i in I. Set X = Π Xi, the Cartesian product of the sets Xi. For every i in I, we have a canonical projection pi : XXi. The product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.

So, am I to understand that for a topology to have a product topology, it has to be for a Cartesian product of topological spaces for index sets? Isn't it a little odd to have index sets that are not indexing something? PDBailey (talk) 03:32, 22 January 2009 (UTC)

Looks fine to me. There is one index set, I which indexes the spaces Xi . Indeed Jmath666 (talk) 04:08, 22 January 2009 (UTC)

Product space page

The search "product space" directs here. Is the any way that there could be a disambiguation between this page and this other Product Space (http://en.wikipedia.org/wiki/The_Product_Space)?

Thanks!

18.111.59.229 (talk) 21:51, 10 May 2012 (UTC)