Topology - Boundary of a ball without a point

[physics1000]

TL;DR Summary

Let us say we have f analytic in Ball_1(0).
which means, radius 1, starting at z_0 = 0 point.
If I want to find the boundary of Ball_1(0).
Will the boundary be {0} or {empty}?
Not homework, just an intuition to understand f(z)=1/z function ( for example ) better.

Let us say we have f analytic in ##Ball_1(0)##. which means, radius 1, starting at ##z_0 = 0## point. If I want to find the boundary of ##Ball_1(0)##. Will the boundary be ##{0}## or ##{\emptyset}##? Not homework, just an intuition to understand ##f(z)=\frac 1 z## function ( for example ) better.

 

[mathman]

As stated, assuming real space of n dimensions, the boundary consists of all points at a distance 1 from the origin. Your statement is confusing? What has f to do with the ball boundary?

 

[physics1000]

Hi, it does not matter anymore.
Asked someone from my course which I trust and he said to me the answer :)
Basically, As I said.
At complex analysis.
If you have the Ball I said, with radius 1 and it beginning at point zero.
If you create the set of that ball without the point zero, then the boundary will be the unision of ##|z|=1## and ##|z|=0##
Thanks though

 

[wrobel]

I suspect that "someone which you trust" will get the same unsatisfactory mark at an exam as you :)

 

[AndreasC]

I suspect that "someone which you trust" will get the same unsatisfactory mark at an exam as you :)

They're not wrong though, op just phrased it confusingly. Op is asking if you remove the point 0 from the ball, is it now a boundary point? The answer is yes it is.

 

[WWGD]

In a metric space, the boundary of a set S is the set of points at distance 0 from the set.