# In topology: homeomorphism v. monotone function

• jjou
In summary, the conversation discusses proving that a bijection is a homeomorphism if and only if it is a monotone function. The conversation goes on to discuss using the intermediate value theorem and the fact that if f is not monotone, then there exists x and y where f(x)=f(y). With these suggestions, the proof is obtained.
jjou
1. Let $$f:\mathbb{R}\rightarrow\mathbb{R}$$ be a bijection. Prove that f is a homeomorphism iff f is a monotone function.

I think I have it one way (if f is monotone, it is a homeomorphism), but I'm stuck on the other way (if f is a homeomorphism, then it is monotone). I tried to prove this using contradiction.

Assume it is a homeomorphism but not monotone. Then there exists a,b,c in R s.t. a<b and a<c but f(a)<f(b) and f(a)>f(c). I think the statement I want to eventually contradict is the following: $$U\subset\mathbb{R}$$ is open iff $$f(U)\subset\mathbb{R}$$ is open.

Could someone give me a small hint? Thanks. :)

Use the intermediate value theorem (draw a sketch!).

Can you prove that if f is not monotone, then f(x)= f(y) for some $x\ne y$?

Thanks! I got it very quickly using both of your suggestions. :D

## 1. What is a homeomorphism?

A homeomorphism is a type of function in topology that preserves the topological structure of a space. It is a continuous function that has a continuous inverse, meaning that both the function and its inverse map open sets to open sets.

## 2. What is a monotone function?

A monotone function is a type of function that preserves the order of elements in a given set. It is either non-decreasing (also known as monotone increasing) or non-increasing (monotone decreasing). This means that the function either preserves or reverses the ordering of elements in the set.

## 3. What is the difference between a homeomorphism and a monotone function?

While both homeomorphisms and monotone functions preserve certain properties of a space, they differ in the specific properties that they preserve. A homeomorphism preserves the topological structure of a space, while a monotone function preserves the order of elements in a set.

## 4. Can a homeomorphism also be a monotone function?

Yes, a homeomorphism can also be a monotone function. However, not all homeomorphisms are monotone functions and vice versa. Whether or not a function is both a homeomorphism and a monotone function depends on the specific properties of the space and the function itself.

## 5. How are homeomorphisms and monotone functions used in topology?

Homeomorphisms and monotone functions are important tools in topology for studying and understanding the properties of spaces. They allow for the comparison and classification of different spaces based on their topological and ordering properties.

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