Question

Let (X,d) be a metric space and \[x\in X\], show that \[\left\{ y\in X|d(y,x)>r \right\}\]is open for any\[r \in \mathbb{R} \]

Answer 1

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Let (X,d) be a metric space and \[x\in X\], show that \[\left\{ y\in X|d(y,x)>r \right\}\]is open for any\[r \in \mathbb{R} \]

Answer 2

huh?

Answer 3

@ikram002p I think you're the only qualified one for this.

Answer 4

Oh, hmm. This question is pretty intuitive actually. Here is what Wikipedia says:
\[B(x;r) = \{y\in M:d(x,y) < r\}\]Explicitly, a subset U of M is called open if for every x in U there exists an r > 0 such that B(x;r) is contained in U.

This is all I know about metric spaces but I'm going to try to help you since you paid for this and Ikram isn't here yet.

Answer 5

Oh, so you're asking if the set you posted is an "open" set in that way?
And ikram is here so I guess I'll leave.

Answer 6

Could you send me the link for that site?

Answer 7

https://en.wikipedia.org/wiki/Metric_space#Open_and_closed_sets.2C_topology_and_convergence

Answer 8

ok i'm here now :D.
let me conjuring my mind up...

Answer 9

I guess it makes sense because the set is a part of the standard topology on \(\mathbb R^2\) haha.
I'll stop.

Answer 10

@Tommynaiter you need to show these stuff (familiar with them?)

\(\Large \tau=\left\{ y\in X|d(y,x)>r \right\}\)
the trivial step i- \(\varnothing,X \in \tau \)
ii- \(\Large U \bigcap V \in \tau ,for ~~U,V\in \tau\)
iii-\(\Large \cup_{\alpha \in \Delta} V_\alpha \in \tau \) for an indexed family \(\ v_\alpha \in \tau\)

Answer 11

also i need you to write down your text book definition for a metric space... the one on wiki is bullsh!t.

Answer 12

brb...

Answer 13

I know some of those, but that does not look familiar at all. Definition:
Let (X,d) be a metric space, let \[x_0\in X\] and let \[r>0\] be a positive reel number. Then the set \[\left\{ x \in X|d(x,x_0)<r \right\}\] consisting of the points X which distance to x_0 is less than r, is called the open ball, with center in x_0 and radius r. It is defined: \[B_r(x_0)=\left\{ x \in X|d(x,x_0)<r \right\}\]

Answer 14

Waaait, so we really are talking about topological spaces here?
Wow that definition is very close to what I've seen.

Answer 15

Yes, that is correct, topological spaces.

Answer 16

D'oh, I could have helped you here then.

Answer 17

Hmm yea, the definition from my book, looks a lot like the one you found on wikipedia @ParthKohli

Answer 18

@ganeshie8 Do you know anything about prooving something like this? This is Math-analysis 1, and you've helped med with math-analysis 0 before

Answer 19

See if this helps
http://math.stackexchange.com/questions/1634312/topology-open-set

Answer 20

That sure does! Thank you :)

Answer 21

Np :)