Gluing points of [0, 1] to get [0, 1]^2

[GridironCPJ]

By Peano's space-filling curve, there exists a continuous map f: I -> I^2 whos image fills up the entire square I^2 (where I=[0, 1]). This can also be represented by gluing points of I together. Which points of I get glued together? I was looking at the proof of Peano's space-filling curve and I kind of get the idea it's all of them, although I'm having trouble justifying that. Would anyone like to shed some light on this?

 

[lavinia]

This can also be represented by gluing points of I together. Which points of I get glued together? I was looking at the proof of Peano's space-filling curve and I kind of get the idea it's all of them, although I'm having trouble justifying that.

Can you explain what you mean by this?

 

[Jamma]

Hmm, how about you glue together any two points x,y whenever f(x)=f(y)? The map f applied to this quotient space will then be a bijection.

The universal property of quotients will mean that the induced map is continuous and of course a continuous bijection from a compact space to a Hausdorff one is a homeomorphism, so we are done :)

 

[Jamma]

Of course, another way you can easily see this is to just think that the square is the image of the line, but where overlapping points on the line are the same.

 

[lavinia]

Hmm, how about you glue together any two points x,y whenever f(x)=f(y)? The map f applied to this quotient space will then be a bijection.

The universal property of quotients will mean that the induced map is continuous and of course a continuous bijection from a compact space to a Hausdorff one is a homeomorphism, so we are done :)

cool.

So the square is a quotient space of the interval.

 

[lavinia]

By Peano's space-filling curve, there exists a continuous map f: I -> I^2 whos image fills up the entire square I^2 (where I=[0, 1]). This can also be represented by gluing points of I together. Which points of I get glued together? I was looking at the proof of Peano's space-filling curve and I kind of get the idea it's all of them, although I'm having trouble justifying that. Would anyone like to shed some light on this?

I think but am not sure that space filling curves are the uniform limits of continuous functions on closed intervals. A theorem states that this limit is itself continuous.

These functions are those maze like curves. They are designed - I think - to approach arbitrarily closely to every point in the square.

This trick applies to other sequences of uniformly continuous functions to produce other weird limits such as the Devil's staircase,