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

1. Jun 1, 2012

### 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?

2. Jun 1, 2012

### lavinia

Can you explain what you mean by this?

3. Jun 1, 2012

### 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 :)

Last edited: Jun 1, 2012
4. Jun 1, 2012

### 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.

5. Jun 1, 2012

### lavinia

cool.

So the square is a quotient space of the interval.

6. Jun 1, 2012

### lavinia

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 wierd limits such as the Devil's staircase,