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

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In summary, according to the space-filling curve, there exists a continuous map from the interval I to the square I^2 where all points get glued together. This can be represented by gluing points of I together.
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GridironCPJ
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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?
 
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  • #2
GridironCPJ said:
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?
 
  • #3
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 :)
 
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  • #4
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
Jamma said:
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.
 
  • #6
GridironCPJ said:
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,
 

1. What does it mean to "glue points" in mathematics?

Gluing points in mathematics refers to a technique used to combine multiple objects into a single object by identifying certain points on the objects as equivalent. This creates a new object that has the same structure as the individual objects, but with the identified points "glued" together.

2. How is the gluing technique used to get [0, 1]^2 from [0, 1]?

In this case, the gluing technique involves identifying the endpoints of the interval [0, 1] as equivalent points. This creates a new object, [0, 1]^2, which is a square with sides of length 1. The corners of the square are identified with the endpoints of the interval [0, 1], and the sides of the square are identified with the points in between.

3. Why is gluing points of [0, 1] useful in mathematics?

Gluing points allows mathematicians to create new and more complex objects from simpler ones. It also helps to better understand the structure and properties of mathematical objects by considering how they are related to other objects through the gluing process.

4. Are there any other examples of gluing points in mathematics?

Yes, gluing points is a common technique in topology and geometry. For example, gluing points on the boundary of a disk can create a sphere, or gluing points on the edges of a rectangle can create a torus. This technique can also be used to create more abstract objects, such as the projective plane.

5. How does gluing points of [0, 1] relate to other mathematical concepts?

Gluing points is closely related to the concept of equivalence relations in mathematics. By identifying certain points as equivalent, we are essentially creating an equivalence relation on the original object. This technique is also important in understanding the fundamental group and homotopy theory in topology.

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