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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeMay 24th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI created [[Cantor space]] to record its definition as a locale, but goodness knows there is no end to what might be written about it.

   I created Cantor space to record its definition as a locale, but goodness knows there is no end to what might be written about it.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 24th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI cross-linked [[Cantor space]] from (newly created) Examples-sections at [[topological space]] and [[locale]] and [[topology - contents]] Eventually hopefully this sidebar for topology is expanded to something that reflects the scope of the relevant nLab articles

   I cross-linked Cantor space from (newly created) Examples-sections at topological space and locale and topology - contents

   Eventually hopefully this sidebar for topology is expanded to something that reflects the scope of the relevant nLab articles

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 21st 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added a bit more to [[Cantor space]], including the abstract characterization up to homeomorphism (which was oddly missing, since what was there seemed to be leading right up to that point). While I was at it, I created [[perfect space]] (with [[perfect set]] redirected to it).

   I added a bit more to Cantor space, including the abstract characterization up to homeomorphism (which was oddly missing, since what was there seemed to be leading right up to that point). While I was at it, I created perfect space (with perfect set redirected to it).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 8th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexI gave _[[Cantor space]]_ more of an Idea-section. Then I expanded the discussion at _[As a subspace of the real line](https://ncatlab.org/nlab/show/Cantor+space#AsSubspaceOfTheRealLine)_ with full detail. The same discussion I also copied over to _[[Tychonoff product]]_ in [this example](https://ncatlab.org/nlab/show/Tychonoff+product#CantorSpace).

   I gave Cantor space more of an Idea-section. Then I expanded the discussion at As a subspace of the real line with full detail. The same discussion I also copied over to Tychonoff product in this example.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 8th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI'm interested in the last sentence of the Idea section. In what way is Cantor space used to construct the Peano curve?

   I’m interested in the last sentence of the Idea section. In what way is Cantor space used to construct the Peano curve?

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 8th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWorth adding the coalgebraic description? What is it, the terminal coalgebra for the endofunctor on $Top$, $X \mapsto X + X$?

   Worth adding the coalgebraic description? What is it, the terminal coalgebra for the endofunctor on $Top$, $X \mapsto X + X$?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 8th 2017
   • (edited May 8th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexTodd, sorry, I should have been more explicit. Maybe I should write "may be used to neatly organize the construction" rather than "may be used to construct": I am thinking of picking a continuous surjection $Cantor \overset{t}{\to} Cantor \times Cantor$ (e.g. unshuffle), then observing that there is easily a continuous surjection $s \colon Cantor \to [0,1]$ and, with a tad more work, that every continuous function from $Cantor$ to a linear space may be extended along the defining embedding $Cantor \hookrightarrow [0,1]$ (by linear interpolation). Then applying this extension to the surjection $Cantor \to Cantor \overset{t}{\times} Cantor \overset{(s,s)}{\to} [0,1] \times [0,1]$ gives the desired continuous surjection.

   Todd, sorry, I should have been more explicit. Maybe I should write “may be used to neatly organize the construction” rather than “may be used to construct”: I am thinking of picking a continuous surjection $Cantor \overset{t}{\to} Cantor \times Cantor$ (e.g. unshuffle), then observing that there is easily a continuous surjection $s \colon Cantor \to [0,1]$ and, with a tad more work, that every continuous function from $Cantor$ to a linear space may be extended along the defining embedding $Cantor \hookrightarrow [0,1]$ (by linear interpolation). Then applying this extension to the surjection $Cantor \to Cantor \overset{t}{\times} Cantor \overset{(s,s)}{\to} [0,1] \times [0,1]$ gives the desired continuous surjection.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 8th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexWell, I'll be. That's rather nice, Urs. Never saw that before (and see nothing wrong with it). David: that's right.

   Well, I’ll be. That’s rather nice, Urs. Never saw that before (and see nothing wrong with it).

   David: that’s right.

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 8th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOk, I'll put it in.

   Ok, I’ll put it in.

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 9th 2017
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexBeing much taken with the simplicity of this Peano curve as sketched by Urs #7, I looked around and saw this is called the Lebesgue space-filling curve, which has another nice properties such as being differentiable almost everywhere. It's obviously similar in flavor to the [Cantor-Lebesgue function](https://en.wikipedia.org/wiki/Cantor_function). Anyway, I went ahead and bashed out the construction at [[Peano curve]], with a proof of continuity. It was just a quick and dirty job, which I may see about cleaning up later. Not many cross-links were inserted. (It's now bedtime for me.)

    Being much taken with the simplicity of this Peano curve as sketched by Urs #7, I looked around and saw this is called the Lebesgue space-filling curve, which has another nice properties such as being differentiable almost everywhere. It’s obviously similar in flavor to the Cantor-Lebesgue function.

    Anyway, I went ahead and bashed out the construction at Peano curve, with a proof of continuity. It was just a quick and dirty job, which I may see about cleaning up later. Not many cross-links were inserted. (It’s now bedtime for me.)

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks, Todd! That's very nice. I didn't know that Lebesgue's name is associated with this.

    Thanks, Todd! That’s very nice. I didn’t know that Lebesgue’s name is associated with this.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 9th 2017
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexI did some more jiggering with [[Peano curve]], which then led me to add to [[Cantor space]] a proof of the Hausdorff-Alexandroff theorem, which says that every compact metric space is a continuous image of Cantor space.

    I did some more jiggering with Peano curve, which then led me to add to Cantor space a proof of the Hausdorff-Alexandroff theorem, which says that every compact metric space is a continuous image of Cantor space.