On a very fundamental level, how is a plane or a volume any different than a line if they are all one-to-one? What, mathematically, separates them, other than their obvious superficial differences?
I mean, I’m not a maroon… I get that a plane is defined by any 3 non-colinear points and a volume is defined by any 4 non-planar points. And I get visually that they are very different things. But if they are all 1-to-1 with each other, aren’t they all just different names for the same “thing,” in some way?
I don’t even know what I’m trying to ask I guess because I do understand the differences, but it bothers me that they are 1-to-1 but still so different from one another practically.
I don’t know what you mean, but they aren’t all the same. A line and a plane are infinite while a volume defines a finite space. Circles and squares have more of a relationship to a volume.
You can establish a one-to-one relationship between the points on a line and the points on a plane, but it isn’t a continuous function. In other words, with points close to each other on the line, the corresponding points on the plane need not be close. In practical terms, the line and plane don’t have the same structure. (A mathematician would talk about them having different topologies.)
What OP is talking about is that lines, planes, and spaces have the same order of infinity – for instance, there is a one to one correspondence between the points in a square and the points on one of its edges. Unfortunately, I don’t really know math, so I can’t explain it.
I was perhaps using the word volume incorrectly. I did not mean to imply a finite space.
Giles, thank you for your insights. The function to map points on a line to a plane not being continuous helps me see a “fundamental” difference between them.
What about the Peano or Hilbert space-filling curves? For that matter, what about a function which uses decimal representations to map pairs to reals:
(… ABC.DEF…, … abc.def…) –> (… AaBbCc.DdEeFf…)
(There’s a question-mark on these “what abouts” because my own math skill is no longer adequate for confidence of the answers, if it ever was.)
To the OP: The prime numbers are one-to-one with the rational numbers. That is easier to understand than plane–>line, but still paradoxical enough to amuse.
Suppose you’re interested in classifying spaces up to bijective (= 1-1) functions f:X -> Y between spaces X, Y. If you put no restrictions on f, so that any function of sets works, then you don’t wind up with a very useful notion. As you said, there’s a bijection between R and R^2. That’s not necessarily (and usually isn’t, for exactly the reason you mentioned) the right category of functions to take. Topologists, for example, are interested in the classification that results when you require that f and its inverse be continuous (called a homeomorphism). The spaces R and R^2 are no longer the same in that case: deleting a point p from R makes the result disconnected, while deleting f§ from R^2 is still connected. With a bit more machinery, you can extend that to prove that R^n and R^m are not homeomorphic for n, m distinct.
But that’s not necessarily the only relation to care about either; topologists, for example, are also interested in a more complicated notion called homotopy equivalence. Here R^n and R^m happen to be homotopy equivalent for any n, m, but the classification for more interesting spaces is more interesting. You can also require that f and its inverse be not just continuous, but actually smooth. If the objects you’re looking at are groups, you want f and its inverse to preserve that group structure. And so forth.
The point of these different classifications is that they’re genuinely different and give you different math. If you want to talk about area and volume, for example, then you probably want at least continuous and probably smooth maps. Again, if you’re a topologist, then you probably care about more exotic notions for various awesome but technical reasons; see exotic spheres, for example. In short, you often care more about the objects in question than just their properties as sets.
This point about homeomorphism is especially elucidating for me. So basically a line is quite different from a plane because even though you can establish a one-to-one correspondence between them, the function is not going to be continuous and therefore you aren’t going to be able to ever “transform” a line into a plane by some sort of mapping. Is that essentially correct?
ETA: Posted before reading the previous post which seems to be answering my question exactly.
Very cool stuff. Thank you so much for your info and your link to exotic spheres. I was a physics major so I am fascinated by a lot of upper mathematics, and I never really got any topology other than what I’ve studied myself or what little introduction you get during regular mathematics education.
Now I feel I understand better what “fundamentally” separates a line from a plane, and also even a sphere from a torus, like you say.
Oh but a space filling curve can be continuous. The operative phrase is a “non-self-intersecting space filling curve cannot be continuous”. Omit the part about non-self-intersecting and the claim is wrong. Dimension theory is the crucial clue. And, just as you imagine, a line segment has dimension 1, a square has dimension 2, and a cube dimension 3. And dimension is a topological invariant.
