Proving the Existence of Fixed Points in Compact Metric Spaces

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In summary, the conversation is about proving that for every continuous function f:X->X of a metric and compact space X, which satisfy for each two different x and y in X p(f(x),f(y))<p(x,y) where p is the metric on X, there's a fixed point, i.e there exist x0 s.t f(x0)=x0. The conversation includes discussing the Banach contraction mapping theorem and using Cauchy sequences to prove the claim. The conversation also touches on determining whether certain spaces satisfy S2 or Sep, which stand for second countable and separable, respectively. The conversation ends with discussing possible approaches to proving separability for C^k[0,1] and l_2
  • #1
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I need to prove that for every continuous function f:X->X of a metric and compact space X, which satisfy for each two different x and y in X p(f(x),f(y))<p(x,y) where p is the metric on X, there's a fixed point, i.e there exist x0 s.t f(x0)=x0.

obviously i thought assuming there isn't such a point i.e that for every x in X f(x)!=x
now because X is compact and it's a metric space it's equivalent to sequence compactness, i.e that for every sequence of X there exist a subsequence of it that converges to x0.

now [tex]p(f(x_{n_k}),x_{n_k})[/tex], because they are not equal then there exist e0 such that: [tex]p(f(x_{n_k}),x_{n_k})>=e0[/tex]
now if x_n_k=f(y_n_k) y_n_k!=x_n_k, we can write it as:
p(x_n_k,x0)+p(x0,y_n_k)>=p(x_n_k,y_n_k)>e0
now if y_n_k were converging to x0, it will be easier, not sure how to procceed...

what do you think?
 
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  • #2
This is a special case of the Banach contraction mapping theorem. A proof would go as follows: Let x0 be any point in X, and let xn=f(xn-1) for n>1. Claim: {x_n} converges.

Post back if you need more hints.
 
  • #3
well cauchy sequnce obviously will do here.
[tex]p(f(x_n),f(x_{n_k}))<p(x_n,x_n_k)=p(f(x_{n-1}),f(x_{n_k-1}))<p(x_{n-1},x_{n_k-1})<...<p(x_0,x_{n_k-n})[/tex]
now if x_{n_k-n} converges to x_0 (which can be assumed cause it's compact and metric), it will be easy to prove your claim, cause then for every e>0 s.t k is big enough:
p(x_0,x_{n_k-1})<e/2 and also p(x0,x_{n_k-n})<e/2
so p(x_n-1,x0)<=p(f(x_n),f(x_n_k))+p(f(x_n_k),x0)<...<e.

is this wrong? I have a sneaky suspicion that yes.
 
  • #4
If n > n_k, then [itex]x_{n_k - n}[/itex] doesn't make sense. Also I don't see how you can assume that "x_{n_k-n} converges to x_0", because it isn't true.

You started out with the right idea. Let n>m, and consider p(xn, xm). Show that we can make this arbitrarily small. This would imply that {xn} is Cauchy and hence convergent (because X is compact). Then we can use the continuity of f to conclude that f has a fixed point (how?).
 
  • #5
well, p(xn,xm)<p(xn-1,xm-1)<...<p(xn-m,x0)
now how do i procceed from here?
I mean if I assume n-m is big enough, s.t x_n-m->x0 then that will do, not sure that this is correct...
 
  • #6
I have another two questions, I need to answer if the next spaces satisfy S2 or Sep, the spaces are with they metrics affiliated with them, in here:
http://www.math.tau.ac.il/~shustin/course/tar5top.xet.pdf [Broken]
in questions 4,5 (disregard the herbew words near them) there listed the spaces.

well what i think is that because if a space is metric and it satisfies S2 then it also satisifes S2, and always when S2 is satisifes then also Sep is satisifed, then it's easy to check fo Sep, i think it follows that for the first the space follows both of them, while in the second it doesn't satisify either of them.

not sure how argue that?
I mean can I find a countable basis for the C^k[0,1]?
or a countable dense set in it?
what do you think?
 
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  • #7
loop quantum gravity said:
well, p(xn,xm)<p(xn-1,xm-1)<...<p(xn-m,x0)
now how do i procceed from here?
I mean if I assume n-m is big enough, s.t x_n-m->x0 then that will do, not sure that this is correct...
To finish off, you can use the triangle inequality
[tex]\rho(x_{n-m},x_0) < \rho(x_{n-m},x_{n-m-1}) + \rho(x_{n-m-1}, x_{n-m-2}) + \cdots + \rho(x_1, x_0)[/tex]
coupled with the observation that
[tex]\rho(x_n, x_{n-1}) < \rho(x_1, x_0)[/tex].
 
  • #8
As for your other question: I'm guessing S_2 means second countable (has a countable basis) and Sep means separable (has a countable dense subset), right?

And you have the right idea: a metric space is separable iff it's second countable. I would use separability here. For C^k[0,1], try to see if Weierstrauss's theorem is helpful. For l_2, I would think about the subspace consisting of sequences with only finitely many terms. This is certainly dense in l_2, but is it countable? No. So how about we restrict these sequences to those with rational terms?
 
  • #9
it seems eventually that munkres has a similar questions with hints which were helpful.
 

What is compactness?

Compactness is a mathematical concept that describes the property of a set being small or finite in some sense. In topology, a compact set is one that is closed and bounded, meaning it contains all its limit points and can be covered by a finite number of open sets.

What is continuity?

Continuity is a mathematical concept that describes the property of a function being connected or unbroken. In topology, continuity means that small changes in the input of a function result in small changes in the output. This is often visualized as a graph with no breaks or gaps.

How are compactness and continuity related?

Compactness and continuity are closely related in topology. A continuous function on a compact set will preserve the compactness of that set, meaning the image of the set under the function will also be compact. Similarly, a compact set can be characterized as one that is both closed and bounded, and continuous functions preserve both these properties.

What are some real-life examples of compactness and continuity?

Compactness and continuity have many real-life applications, including in physics, engineering, and economics. For example, the concept of compactness is used to describe the behavior of particles in a confined space, while continuity is essential for modeling the flow of fluids or electricity.

What are some common techniques for proving compactness and continuity?

There are many different techniques for proving compactness and continuity, depending on the specific context and problem at hand. Some common techniques include using the definition of compactness and continuity, using topological properties such as connectedness and compactness, or using mathematical theorems such as the Heine-Cantor theorem or the Bolzano-Weierstrass theorem.

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