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!:tongue:

 

[TheShrike]

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

Suddenly it all makes sense.