Talk:Kolmogorov space

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. --Cronholm144 03:34, 16 June 2007 (UTC)

I have a problem with this paragraph:

Now consider a structure that can be placed on topological spaces, such as a metric. We can define a new structure on topological spaces by letting an example of the structure on X be simply a metric on KQ(X). This is a sensible structure on X; it is a pseudometric. (Again, there is a more direct definition of pseudometric.)

I don't think of a metric as a structure that I put on a topological space, but rather as a structure on a set that defines a topology. In this view, one could define metrizable topological spaces as those that come from a metric, and pseudo-metrizable topological spaces as those whose KQ comes from a metric. AxelBoldt

Either way of thinking about things is valid. But placing structures on a topological space is more general (or can be, depending on how you define structure), and almost any structure used in practice can be transferred this way, whether it uniquely defines the topology or not. And saying that the KQ of a pseudometric space is a metric space is certainly saying more than that the KQ of a pseudometrisable space is a metrisable space; the KQ doesn't just give you the property of being pseudometrisable but the structure of having a specific pseudometric. IOW, L²(R) is not only metrisable; there is a unique best metric on it (from the POV of the pseudometric that we started from, of course). Conversely, if KQ(X) is a metric space, then X is not only pseudometrisable; there is a unique best pseudometric on X (again from the POV of the metric that we began with on KQ(X).)

(This reminds me that I should write Structure (mathematics) one of these days. The trick is to keep the talk at a low level, with lots of examples, while keeping in mind that at the very end of the article, I'll mention faithful forgetful functors.)

Thanks for the copyedits, BTW!

— Toby 02:10 Aug 12, 2002 (PDT)

Contents 1 Sierpinski space 2 Kolmogorov quotient 3 Examples? 4 Topologically distinguishable

[edit] Sierpinski space

The article on Sierpinski space states that:

A topological space is Kolmogorov if and only if it is homeomorphic to a subspace of a power of the Sierpinski space S.

Curiously, this article does not make this claim; it seems like an important claim to me. linas 17:53, 20 November 2005 (UTC)

[edit] Kolmogorov quotient

Does anybody happen to know where the terminology "Kolmogorov quotient" comes from? I've tried several google searches: [1], [2], [3] (in the last one I tried to get rid of various wikipedia mirrors and similar sites). BTW if the T₀-spaces are taken as the reflective subcategory of the category Top then the Kolmogorov quoetient as described in this entry is precisely the T₀-reflection. --Kompik 13:40, 16 June 2006 (UTC)

[edit] Examples?

Can someone knowledgable provide some appropriate (perhaps canonical) examples of spaces which both are and are not T₀?

Er, probably you meant "... both spaces which are, and spaces which are not, T₀". Any Hausdorff space -- say, the real line -- is T₀; any indiscrete space with more than one point is not. Maybe you want a more elaborate example? JadeNB 03:47, 2 January 2007 (UTC)

[edit] Topologically distinguishable

Does the second sentence in this paragraph:

Otherwise, x and y are said to be topologically distinguishable. Loosely speaking, this means that the topology on X is capable of distinguishing between x and y.

really add anything? It seems to be just excess verbiage. JadeNB 03:47, 2 January 2007 (UTC)