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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.
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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
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The 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
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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.
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Someone changed "classifying space" to "classifying topos" throughout. A space is not a topos, so this is just false, as stated. It
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