Difference between open sphere and epsilon-neighbourhood - Metric Spaces

[TheShrike]

In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows:

An open sphere S(x_0,r) in a metric space R (with metric function \rho(x,y)) is the set of all points x\in R satisfying \rho(x,x_0)<r. The fixed point x_0 is called the center; the number r is called the radius.

An ε-neighbourhood of the point x, denoted O(x,\epsilon), is an open sphere of radius ε and center x_0.

How is the ε-neighbourhood a significant definition? It seems to be just the open sphere with a different radius symbol. If we have a neighbourhood of a point x (as per the definition) does this x have to lie within the open sphere? I mean, I assume it must, but this doesn't seem to be captured by the definition. What am I missing?

Thanks.

 

[HallsofIvy]

In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows:

An open sphere S(x_0,r) in a metric space R (with metric function \rho(x,y)) is the set of all points x\in R satisfying \rho(x,x_0)<r. The fixed point x_0 is called the center; the number r is called the radius.

An ε-neighbourhood of the point x, denoted O(x,\epsilon), is an open sphere of radius ε and center x_0.

No. an \epsilon-neighborhood of the point x is open sphere with radius \epsilon and center x, not some x_0 (that wouldn't make sense because x_0 is not mentioned in the notation "O(x, \epsilon)").

How is the ε-neighbourhood a significant definition? It seems to be just the open sphere with a different radius symbol. If we have a neighbourhood of a point x (as per the definition) does this x have to lie within the open sphere? I mean, I assume it must, but this doesn't seem to be captured by the definition. What am I missing?

Thanks.

An "\epsilon-neighborhood of x" is specifically the open ball of radius \epsilon with radius \epsilon that is centered at x.

 

[Studiot]

An ε-neighbourhood of the point x, denoted O(x,ϵ), is an open sphere of radius ε and center x0.

I think this should read

An ε-neighbourhood of the point x, denoted O(x,ϵ), is an open sphere of radius ε and center x.

I have bolded the difference.

I prefer the term open ball to open sphere however.

edit I see Halls of ivy just beat me.

 

[HallsofIvy]

I think this should read

An ε-neighbourhood of the point x, denoted O(x,ϵ), is an open sphere of radius ε and center x.

I have bolded the difference.

I prefer the term open ball to open sphere however.

edit I see Halls of ivy just beat me.

Yahoo!

 

[TheShrike]

That's as I suspected. There must be a typo in the book.

Suddenly it all makes sense.