Question

i have another question in metric topology.

Answer 1

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

Answer 2

@oldrin.bataku

Answer 3

@zzr0ck3r

Answer 4

@zzr0ck3r

Answer 5

I cant see the full question

Answer 6

describe the .....

It goes off the side of the page.

Answer 7

describe the set \[B _{0}( (0,0);1)

Answer 8

\(\{x\in \mathbb{R}^2 \mid \sqrt{x_1^2+x_2^2} <1\ \}\)

Answer 9

It is a circle around the origin of radius 1. We include everything in the circle, but not the "border" of the circle

Answer 10

2) describe the open ball \[B _{0}( (0,0);1)\]

Answer 11

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 \(1\)

Answer 12

Er the inequality should read less than

Answer 13

<1 @oldrin.bataku

Answer 14

word*

Answer 15

I'm on my phone and OpenStudy lags

Answer 16

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

Answer 17

so, what is the description?

Answer 18

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

Answer 19

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.

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

ok . thank you sirs

Answer 24

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

Answer 25

do you know what a homeomorphism is?

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

its topological space question

Answer 30

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

Answer 31

any easy one will be ok

Answer 32

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

Answer 33

Cauchy definition

Answer 34

Which is?

Answer 35

@zzr0ck3r i do not know how to prove it

Answer 36

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

Answer 37

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

Answer 38

no

Answer 39

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?

Answer 40

yes

Answer 41

so , that is the prove

Answer 42

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 \(\mathbb{R}\).

Answer 43

Do you need to show they are continuous?

Answer 44

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

Answer 45

just go from the known fact that they both are

Answer 46

Then you are done.

Answer 47

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

Answer 48

waw

Answer 49

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

Answer 50

i mean you are a genius

Answer 51

not at all...

Answer 52

i wish i am good like you

Answer 53

keep at it.

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

Answer 54

1 year ago*

Answer 55

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}

Answer 56

?

Answer 57

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.



In this picture, it is all the "white" inside the circle.

Case 2: Mathematical description: \(B_0((0,0), 1)=\{(x_1, x_2)\in \mathbb{R}^2 \mid x_1^2+x_2^2<1\}\)

Answer 58

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.

Answer 59

ok. got it . i have another quation

Answer 60

close this and ask a new one.

Answer 61

ok