Which of the following spaces are homeomorphic?

Question

Let $X$ be a metric space and $f\colon X \rightarrow \mathbb R$ be a continuous function. Let $G = \{ (x , f(x) ) : x \in X \}$ be the graph of $f$. Then which one is true?

$G$ is homeomorphic to $X$
$G$ is homeomorphic to $\mathbb R$
$G$ is homeomorphic to $X \times \mathbb R$
$G$ is homeomorphic to $\mathbb R \times X$

Please help me how to solve this problem.

Thank you

Take an easy space, say $X = [0,1]$, and an easy $f$. Then sketch, and look what cannot be right. — Daniel Fischer, Dec 23 '14 at 10:46
After you solve the question, you might want to take a look at http://math.stackexchange.com/questions/429797/if-f-is-continuous-then-g-is-homeomorphic-to-x. — Ludolila, Dec 23 '14 at 11:07

score 5 · Answer 1 · answered Dec 23 '14 at 11:07

HINT: After working out an example as in Daniel Fischer's comment, study the following two natural maps:

$\phi:X\rightarrow G$ given by $\phi(x)=(x,f(x))$,
$\varphi:G\rightarrow X$ given by $\varphi(x,f(x))=x$.

In particular study their continuity and compute their compositions.

Learnmore · Answer 2 · 2015-01-25T07:27:48.160

2

Consider $X=(0,1)$ and $f(x)=x$ then $G=\{(x,x):x\in (0,1)\}$ which is a closed set but $(0,1)$ is not

Again Consider $X=[0,1]$ which is compact and $f(x)=x$.Its graph is compact but $\mathbb R$ is not.

NOTE:Consider $f(x)=0$ .Then graph of $f=\{(x,0):x\in \mathbb R\}$ i.e. the $x$ axis .Now remove the point $(0,0)$ from the graph of $f$ ,it becomes disconnected but $\mathbb R^2\setminus \{(0,0)\}$ is not.

edited Jan 25 '15 at 07:27

answered Dec 23 '14 at 11:58

Learnmore

31,062

@ learmoe : please tell me (3) is true or not – user120386 Jan 24 '15 at 17:03
I have edited my answer according to your need @user120386 – Learnmore Jan 25 '15 at 07:28
Be careful - $X=(0.1)$ may not be a closed subset of $\mathbb R$, but as a space itself it is closed. – Dom Jan 25 '15 at 07:33
@ Learner : Thank you so much – user120386 Jan 25 '15 at 11:38
it would be great if you could accept my answer if you think so@user120386 – Learnmore Jan 25 '15 at 11:40

Which of the following spaces are homeomorphic?

2 Answers2