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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeMay 24th 2010

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 24th 2010

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

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeMar 21st 2015

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).

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 8th 2017

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.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeMay 8th 2017

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

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeMay 8th 2017

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

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 8th 2017
• (edited May 8th 2017)

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.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeMay 8th 2017

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

David: that’s right.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeMay 8th 2017

Ok, I’ll put it in.

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeMay 9th 2017

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.)

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMay 9th 2017

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

• CommentRowNumber12.
• CommentAuthorTodd_Trimble
• CommentTimeMay 9th 2017

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.