Why Peano Curve is a curve?
xiaoqian
It's easy to understand that the limit filling the square. But why it is not just a square, but also a curve?
It seems that the limit of a curve sequence is not necessary a curve. Think about this curve sequence: f(n): x*x + y*y = 1/n. (A sequence of concentric circles). The limit of f(n) is just a single point (0, 0). Of course the limit is not a curve, and it is not mean a single point and a curve contain the same number of points. Why Peano Curve means a curve and a sequence contain the same number of points?
Mike Terry
news:618836865.379073.12685...@gallium.mathforum.org...
> Peano Curve, the limit of a curve sequence, is known as "space-filling
curve".
>
> It's easy to understand that the limit filling the square. But why it is
not just a square, but also a curve?
To answer this question, first you have to understand "What is a curve?"
I.e. what is the mathematical definition of a curve that Peano is using?
If you do not understand this, there is no doubt that you will be confused,
and nobody will be able to explain the answer to you, and this thread will
just go on and on aimlessly! (I've seen it before... :-)
So first I will ask you to define:
(...for the purposes of talking about the Peano Curve...) What is the
definition of a curve?
>
> It seems that the limit of a curve sequence is not necessary a curve.
Think about this curve sequence: f(n): x*x + y*y = 1/n. (A sequence of
concentric circles). The limit of f(n) is just a single point (0, 0). Of
course the limit is not a curve, and it is not mean a single point and a
curve contain the same number of points. Why Peano Curve means a curve and a
sequence contain the same number of points?
Well, the above f(n) are not curves! (In the sense we define curves as in
"Peanos Curve"). If you think they are, then you are misunderstanding the
concept of curve - so we should start with you telling us what definition of
Curve you are using, and go from there...
However, even with the correct definition it is the case that the limit of a
sequence of curves is not necessarily a curve. (This is something that must
be shown by some argument for any particular sequence...)
Regards,
Mike.
Arturo Magidin
> Peano Curve, the limit of a curve sequence, is known as "space-filling curve".
>
> It's easy to understand that the limit filling the square. But why it is not just a square, but also a curve?
What is the definition of "curve"? To see if it is a curve, you need
to have a definition (a *formal* definition) of "curve", and then you
need to see if it satisfies that definition. If it does, then it is a
curve; if it doesn't, then it is not.
There are many different ways to define "curve", depending on what
area of mathematics you are dealing with. For example, in algebraic
geometry, a curve is a variety of dimension 1 (see for example William
Fulton, "Algebraic Curves: An Introduction to Algebraic Geometry",
Addison Wesley, 1969, where "curve" is defined on page 150 (!) that
way).
But in analysis, we generally define a "curve" as a continuous
function with domain the unit interval [0,1]. That is, a "plane curve"
is simply a continuous function f:[0,1] --> R^2.
In *that* sense, Peano's curve *is* a curve, because it *is* a
continuous function with domain [0,1]. So there is no argument about
it: it satisfies the definition, so it *is* a curve.
--
Arturo Magidin