i have another question in metric topology.

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i have another question in metric topology.

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

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I cant see the full question
describe the ..... It goes off the side of the page.
describe the set \[B _{0}( (0,0);1)
\(\{x\in \mathbb{R}^2 \mid \sqrt{x_1^2+x_2^2} <1\ \}\)
It is a circle around the origin of radius 1. We include everything in the circle, but not the "border" of the circle
2) describe the open ball \[B _{0}( (0,0);1)\]
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 \(1\)
Er the inequality should read less than
<1 @oldrin.bataku
word*
I'm on my phone and OpenStudy lags
Okay, it seems open balls of radius \(r\) centered at \(p\) are notated \(B_o(p;r)\)
so, what is the description?
yes, in general open is \(B(a,b)\) and closed is \(B[a,b]\)
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.
you know my book did not explain that but this is just the question.
so, number the answers for me because it is two questions
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
ok . thank you sirs
show that the mapping f;R---.R+ defined by f(x)=e^x is homeomorphism
do you know what a homeomorphism is?
that is bijective funtiom f such that f and f^-1 are both continuous
it's bicontinuous, i.e. it's an invertible map that is continuous in both directions
what definition of continuity are you working with? sequential continuity? the Cauchy definition in terms of \(\epsilon,\delta\)?
its topological space question
yes, but are we dealing with \(\mathbb R,\mathbb R^+\) as metric spaces? there are several equivalent definitions of continuity in this case
any easy one will be ok
you need to tell us which one you are using, it's not a matter of one being easy or not :p
Cauchy definition
Which is?
@zzr0ck3r i do not know how to prove it
If it is a topological question. then I assume we are working with the definition pre image of open sets is open.
For sure it is a bijection. Do you know how to prove that?
no
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?
yes
so , that is the prove
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 \(\mathbb{R}\).
Do you need to show they are continuous?
Or just go from the known fact that they both are?
just go from the known fact that they both are
Then you are done.
You can google how to prove they are continuous, and if I prove it, it will look exactly like any proof you find online.
waw
what is waw? I see you make that comment before.
i mean you are a genius
not at all...
i wish i am good like you
keep at it. Go back and look at the questions I asked 2 years ago. They are the same questions you are asking
1 year ago*
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}
?
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: \(B_0((0,0), 1)=\{(x_1, x_2)\in \mathbb{R}^2 \mid x_1^2+x_2^2<1\}\)
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.
ok. got it . i have another quation
close this and ask a new one.
ok

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