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1. • CommentRowNumber1.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMay 31st 2017
   • (edited May 31st 2017)
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   Author: IngoBlechschmidt
   Format: MarkdownItex[[open morphism|Our entry on open morphisms of locales]] contains the following claim: "If $f : X \to Y$ is a continuous map of topological spaces, then the induced morphism between locales is open. The converse holds for sober spaces." The "only if" direction is easy to prove, the sought left adjoint $f_!$ to $f^{-1}$ is simply given by calculating the image of open subsets of $X$ under $f$. But I fail to see the converse direction. It is proved in Moerdijk/Mac Lane (Prop. IX.8.5 on page 506) under the assumption that $Y$ is a $T_1$-space. If this was just an oversight, I'll fix our entry accordingly.

   Our entry on open morphisms of locales contains the following claim: “If $f : X \to Y$ is a continuous map of topological spaces, then the induced morphism between locales is open. The converse holds for sober spaces.”

   The “only if” direction is easy to prove, the sought left adjoint $f_!$ to $f^{-1}$ is simply given by calculating the image of open subsets of $X$ under $f$.

   But I fail to see the converse direction. It is proved in Moerdijk/Mac Lane (Prop. IX.8.5 on page 506) under the assumption that $Y$ is a $T_1$-space.

   If this was just an oversight, I’ll fix our entry accordingly.

2. • CommentRowNumber2.
   • CommentAuthorDaniel Luckhardt
   • CommentTimeJun 1st 2017
   • (edited Jun 1st 2017)
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   Author: Daniel Luckhardt
   Format: MarkdownItexIf I read the paragraph correctly, it does only claim that an _open_ continuous map of topological spaces induces a complete Heyting algebra homomorphism, and hence a homomorphism of locales. The claim without the openess requirement should be wrong, take the bijection between two two-point sets, the domain endowed with the discrete topology and the codomain with the Sierpinski topology: The homeomorphism property for Heyting algebras is violated by the implication $\{x\} \to \emptyset$.

   If I read the paragraph correctly, it does only claim that an open continuous map of topological spaces induces a complete Heyting algebra homomorphism, and hence a homomorphism of locales. The claim without the openess requirement should be wrong, take the bijection between two two-point sets, the domain endowed with the discrete topology and the codomain with the Sierpinski topology: The homeomorphism property for Heyting algebras is violated by the implication $\{x\} \to \emptyset$.

3. • CommentRowNumber3.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeJun 1st 2017
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   Author: IngoBlechschmidt
   Format: MarkdownItexHi Daniel! My bad. I forgot the adjective "open" in my post. Yes, open continuous maps induce open morphisms of locales. And continuous maps which are not open do not in general induce open morphisms of locales (as your example shows). But is it true that, if the induced morphism of locales is open, the continuous map we started with it open? Our entry claims that this is the case if both spaces are sober. But in Moerdijk/Mac Lane there is the requirement that $Y$ is even a $T_1$-space (and $X$ can be an arbitrary space).

   Hi Daniel! My bad. I forgot the adjective “open” in my post.

   Yes, open continuous maps induce open morphisms of locales. And continuous maps which are not open do not in general induce open morphisms of locales (as your example shows).

   But is it true that, if the induced morphism of locales is open, the continuous map we started with it open? Our entry claims that this is the case if both spaces are sober. But in Moerdijk/Mac Lane there is the requirement that $Y$ is even a $T_1$-space (and $X$ can be an arbitrary space).

4. • CommentRowNumber4.
   • CommentAuthorDaniel Luckhardt
   • CommentTimeJun 1st 2017
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   Author: Daniel Luckhardt
   Format: MarkdownItexRemark on your reference: It is actually Prop. IX.__7__.5 on page 506

   Remark on your reference: It is actually Prop. IX.7.5 on page 506

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJun 1st 2017
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   Author: Mike Shulman
   Format: MarkdownItexp521 of the Elephant says > ...this is not quite equivalent to the usual notion of openness for continuous maps of spaces; it is implied by the latter, but in the opposite direction it only implies that the set-theoretic image of each open subset of $X$ is "almost open" in $Y$, in the sense that its subclosure is open. Even when $X$ and $Y$ are both sober, this does not suffice to make all such images open (see [292]), but it obviously does so if $Y$ is a $T_D$ space. Reference [292] is "H. Dobbertin, Measurable refinement monoids and applications to distributive semilattices, Heyting algebras, and Stone Spaces, *Math Z.* 187 (1984), 12-31".

   p521 of the Elephant says

   …this is not quite equivalent to the usual notion of openness for continuous maps of spaces; it is implied by the latter, but in the opposite direction it only implies that the set-theoretic image of each open subset of $X$ is “almost open” in $Y$, in the sense that its subclosure is open. Even when $X$ and $Y$ are both sober, this does not suffice to make all such images open (see [292]), but it obviously does so if $Y$ is a $T_D$ space.

   Reference [292] is “H. Dobbertin, Measurable refinement monoids and applications to distributive semilattices, Heyting algebras, and Stone Spaces, Math Z. 187 (1984), 12-31”.

6. • CommentRowNumber6.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeJun 1st 2017
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   Author: IngoBlechschmidt
   Format: MarkdownItexThank you both! (Unfortunately, schemes are in general not $T_D$ spaces, therefore I can't apply the result to my particular situation.) I'll fix our entry in a moment.

   Thank you both! (Unfortunately, schemes are in general not $T_D$ spaces, therefore I can’t apply the result to my particular situation.) I’ll fix our entry in a moment.