Linked Questions

3
votes
3answers
54 views

Is there a line homomorphism to a disk? [duplicate]

This is a question from an interview. I am confused about this problem. I said yes in that interview because I remember something about Hilbert curve(I mean, is there a line that can fill a square ...
42
votes
11answers
15k views

Probability that 3 points in a plane form a triangle

This question was asked in a test and I got it right. The answer key gives $\frac12$. Problem: If 3 distinct points are chosen on a plane, find the probability that they form a triangle. Attempt 1:...
33
votes
3answers
7k views

Is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$

I need a hint. The problem is: is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$ I'm pretty sure that there aren't any, but so far I couldn't find the proof. My best idea so far is ...
21
votes
3answers
2k views

In what way is the Peano curve not one-to-one with $[0,1]^2$?

In discussion about the question Is there a way to represent the interior of a circle with a curve?, it was mentioned that such a curve cannot be one-to-one (because $[0,1]$ is not homeomorphic to $[0,...
30
votes
3answers
761 views

Is the complement of an injective continuous map $\mathbb{R}\to \mathbb{R}^2$ with closed image necessarily disconnected?

I am interested in the following Jordan curve theorem-esque question: Suppose that you are given a continuous, injective map $\gamma: \mathbb{R}\to \mathbb{R}^2$ such that the image is a closed ...
5
votes
4answers
243 views

Prove or Disprove : There exists a continuous bijection from $\mathbb{ R}^2$ to $\mathbb{R} $

This question was asked to me by a mathematics undergraduate to me and I was not able to solve it. So, I am asking it here. Prove or Disprove : There exists a continuous bijection from $\mathbb{ R}^2$...
7
votes
1answer
2k views

Continuous bijection from $\mathbb{R}^n$ to $\mathbb{R}^m$

Is there a continuous bijection from $\mathbb{R}^n$ to $\mathbb{R}^m$, for $n \neq m$? Such a map would not be an open map, since $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic.
9
votes
2answers
437 views

The Lebesgue measure of the boundary of a simply connected domain

Is the Lebesgue measure of the boundary of a simply connected domain in $\mathbb{R}^n$ necessarily 0? Acturally, I want to know the sufficient condition to guarantee the measure of the boundary of a ...
9
votes
2answers
325 views

Function taking on all values thrice

I recently saw the following puzzle somewhere: Find a continuous, surjective function $f:\mathbb R\mapsto\mathbb R$ that takes on each of its values exactly three times. Or, more technically ...
6
votes
4answers
186 views

On the existence of a continuous bijection $f\colon [0,1]\to [0,1]\times [0,1]$

Let $f$ be a continuous function on $[0,1]$ such that $f([0,1])=[0,1]\times[0,1].$ Then show that $f$ is not one-one. Hints will be appreciated.
0
votes
2answers
676 views

Bijection between $\mathbb R$ and $\mathbb R^N$

Hi, could you give me an example of a bijection between $\mathbb R^T$ and $\mathbb R$ with $T$ being a positive integer which is as regular as possible like $C^{\infty}$ or Lipschitz continuous at ...
3
votes
1answer
600 views

Can there be a embedding from the real number line to the plane such it is both closed and bounded?

Just a little clarification on the question: when I say line I am essentially referring to the real number line and the plane being $\mathbb R^2$. I am really new to the notion of embedding; hence, I ...
1
vote
1answer
284 views

Space-filling curves where proximity in 2D implies proximity in 1D globally

Is anyone aware of a space-filling curve where two points "near" each other in 2D are "near" each other when the curve is extended? This relationship doesn't seem to hold with the Hilbert curve: when ...
0
votes
4answers
119 views

Question regarding the existence of a Bijection $f:\Bbb R^2\to \Bbb R$

I have a hard time believing that there can exist a bijection $f:\Bbb R^2\to \Bbb R$. I just cannot get around my intuition that a one-to-one map of a one-dimensional space (or interval) must also be ...
3
votes
0answers
139 views

Can integrals of higher dimension be reduced to 1D integrals by space filling curves?

On the Wikipedia page on space-filling curves I see the following statement: Wiener pointed out in The Fourier Integral and Certain of its Applications that space filling curves could be used to ...

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