## anonymous one year ago i have another question in metric topology.

1. anonymous

let E=R be endowed with the euclidean metric $d {2}(X,Y)= \sum_{k=1}^{2}(Xk -Yk)^{1/2} for all X=(x1,x2),y=(y1,y2)\in R ^{2}.. descibe the set B _{0}(0,0);1). and also (2) discribe the open ball B _{0}( (0,0);1)$

2. anonymous

@oldrin.bataku

3. anonymous

@zzr0ck3r

4. anonymous

@zzr0ck3r

5. zzr0ck3r

I cant see the full question

6. zzr0ck3r

describe the ..... It goes off the side of the page.

7. anonymous

describe the set $B _{0}( (0,0);1) 8. zzr0ck3r $$\{x\in \mathbb{R}^2 \mid \sqrt{x_1^2+x_2^2} <1\ \}$$ 9. zzr0ck3r It is a circle around the origin of radius 1. We include everything in the circle, but not the "border" of the circle 10. anonymous 2) describe the open ball \[B _{0}( (0,0);1)$

11. anonymous

The Euclidean metric is the normal metric you are used to. We are looking at the open set $$\{ (x,y) : x^2 + y^2 \le 1\}$$, which is just the set of things within distance $$1$$ of our point. This is an open disk of radius $$1$$

12. anonymous

Er the inequality should read less than

13. zzr0ck3r

<1 @oldrin.bataku

14. zzr0ck3r

word*

15. anonymous

I'm on my phone and OpenStudy lags

16. anonymous

Okay, it seems open balls of radius $$r$$ centered at $$p$$ are notated $$B_o(p;r)$$

17. anonymous

so, what is the description?

18. zzr0ck3r

yes, in general open is $$B(a,b)$$ and closed is $$B[a,b]$$

19. zzr0ck3r

When we give the answer as a set, that is a description @GIL.ojei this one is very basic. Can you explain exactly what you are having a problem with.

20. anonymous

you know my book did not explain that but this is just the question.

21. anonymous

so, number the answers for me because it is two questions

22. anonymous

I have never seen the closed ball notated as $$B[p;r]$$, only as $$\bar B(p;r),\operatorname{cl}B(p;r),[B(p;r)]$$, but I imagine it looks like $$B_c(p;r)$$ in @GIL.ojei's book

23. anonymous

ok . thank you sirs

24. anonymous

show that the mapping f;R---.R+ defined by f(x)=e^x is homeomorphism

25. anonymous

do you know what a homeomorphism is?

26. anonymous

that is bijective funtiom f such that f and f^-1 are both continuous

27. anonymous

it's bicontinuous, i.e. it's an invertible map that is continuous in both directions

28. anonymous

what definition of continuity are you working with? sequential continuity? the Cauchy definition in terms of $$\epsilon,\delta$$?

29. anonymous

its topological space question

30. anonymous

yes, but are we dealing with $$\mathbb R,\mathbb R^+$$ as metric spaces? there are several equivalent definitions of continuity in this case

31. anonymous

any easy one will be ok

32. anonymous

you need to tell us which one you are using, it's not a matter of one being easy or not :p

33. anonymous

Cauchy definition

34. zzr0ck3r

Which is?

35. anonymous

@zzr0ck3r i do not know how to prove it

36. zzr0ck3r

If it is a topological question. then I assume we are working with the definition pre image of open sets is open.

37. zzr0ck3r

For sure it is a bijection. Do you know how to prove that?

38. anonymous

no

39. zzr0ck3r

onto: Let $$y\in \mathbb{R}^{+}$$, then $$f(\ln(y))= e^{\ln(y)}=y$$. Note that since $$y\in \mathbb{R}^{+}$$ we have that $$\ln(y) \in \mathbb{R}$$. one-to-one. Suppose $$f(x) = f(y)$$. Then $$e^x=e^y$$ and as a result we have $$\ln(e^x)=\ln(e^y)\implies x=y$$. The function is a bijection. Do you follow?

40. anonymous

yes

41. anonymous

so , that is the prove

42. zzr0ck3r

This proves that it is a bijection. We now need to show that $$\ln(x)$$ is continuous on its domain. Also I guess you might want to show that $$e^x$$ is continuous on $$\mathbb{R}$$.

43. zzr0ck3r

Do you need to show they are continuous?

44. zzr0ck3r

Or just go from the known fact that they both are?

45. anonymous

just go from the known fact that they both are

46. zzr0ck3r

Then you are done.

47. zzr0ck3r

You can google how to prove they are continuous, and if I prove it, it will look exactly like any proof you find online.

48. anonymous

waw

49. zzr0ck3r

what is waw? I see you make that comment before.

50. anonymous

i mean you are a genius

51. zzr0ck3r

not at all...

52. anonymous

i wish i am good like you

53. zzr0ck3r

keep at it. Go back and look at the questions I asked 2 years ago. They are the same questions you are asking

54. zzr0ck3r

1 year ago*

55. anonymous

please show they are continuous but before then, the last quation for discribing a set , which is open ball and which is set B_{0}

56. anonymous

?

57. zzr0ck3r

I am not sure if they want a description like you and I think of when we describe things, or a mathematical description which is a term we use to when referring to a set and a relation on that set. Case 1: visual description: Consider a circle at the origin with radius 1, it is all the points inside the circle. |dw:1438991405420:dw| In this picture, it is all the "white" inside the circle. Case 2: Mathematical description: $$B_0((0,0), 1)=\{(x_1, x_2)\in \mathbb{R}^2 \mid x_1^2+x_2^2<1\}$$

58. zzr0ck3r

Here is a good read on showing things are continuous using $$\epsilon-\delta$$ proofs. http://people.bath.ac.uk/sej20/docs/epsilondelta.pdf Let me know if you have any questions.

59. anonymous

ok. got it . i have another quation

60. zzr0ck3r

close this and ask a new one.

61. anonymous

ok