Talk:Sierpiński space
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Weak topology
editThe current version of the article states that:
- If X is a topological space with topology T, then the weak topology on X generated by C(X,S) coincides with T.
Is this true? I would say it holds only for -spaces. The 2-point indiscrete space could be a counterexample. --Kompik 18:56, 23 February 2006 (UTC)
- Surely for an indiscrete space (X,T), the weak topology on X generated by any set of continuous functions on X is still T, since the indiscrete topology is the weakest possible tolopogy on X. Martin Orr 21:43, 24 May 2006 (UTC)
Classifying "topos"
editSomeone changed "classifying space" to "classifying topos" throughout. A space is not a topos, so this is just false, as stated. It is the case that the topos of sheaves on the Sierpinski space is the classifying topos for global subobjects (called open sets by Artin and Grothendieck) in the category of toposes, but this is a very different statement (proved on page 117 of Johnstone's 1977 Topos Theory book) and is not what the proof in the "Continuous Functions on Sierpinski Space" section says. The linked article in the nLab also uses classifying space, not classifying topos, so the confusion seems to originate here, not there. I am going to change things back to "classifying space" and add a statement to that article explaining that one can also consider classifying spaces up to homeomorphism instead of up to homotopy, and link back to the relevant section of this article for the Sierpinski space example. Furby100 (talk) 08:27, 5 April 2017 (UTC)
Subobject classifier
editThe article, in the "Continuous functions to..." section, made the claim "In other words, the Sierpinski space is the subobject classifier of the category of topological spaces." But this is more than a rephrasing of the property that comes before it. A subobject classifier classifies every monomorphism into an object, but the Sierpinski space classifies only open sets, which are far from the only monomorphisms (or even the only subspaces) in the category of topological spaces. I have removed this line. ThrealAEV (talk) 01:18, 29 June 2022 (UTC)