Understanding the Difference Between Open Balls and Neighborhoods

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In summary, open balls and neighborhoods are not exactly the same thing. A neighborhood is generally used to refer to a small open set, while an open ball is defined in a metric space. There are also three different definitions of a neighborhood, including an open ball around x, an open set containing x, and a set with an open subset containing x. These definitions may vary in their use of small or large numbers, but all convey the same idea.
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mynameisfunk
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Are open balls and neighborhoods the exact same thing? If not, could you please shed some light on this for me?
 
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A neighborhood is just an open set. An open ball requires being in a metric space. The word neighborhood is usually used as opposed to just open set because you want to give the impression that the open set is supposed to be a small one, similar to saying let [tex] \epsilon>0[/tex] vs saying let [tex]M>0[/tex]. They both say the exact same thing but one of them indicates we're interested in picking small numbers and one large numbers. It's not a formal definition but just to give the reader some intuition
 
  • #3
There are at least three inequivalent definitions of "neigborhood of x":

1. An open ball around x.
2. An open set that contains x.
3. A set that has an open subset that contains x.
 

1. What is the definition of an open ball?

An open ball is a set of all points that are within a certain distance (radius) from a given point in a metric space. The center of the open ball is the given point and the radius determines the size of the ball.

2. How is an open ball different from a closed ball?

An open ball includes all points within the specified distance from the center point, but does not include the boundary points. A closed ball, on the other hand, includes the boundary points as well.

3. What is the purpose of using open balls in topology?

Open balls are used in topology to define open sets, which are a fundamental concept in topology. Open sets are sets that contain an open ball around each of its points, and they are used to define continuity, convergence, and other important topological properties.

4. How are open balls related to neighborhoods?

Open balls are a type of neighborhood. A neighborhood is a set that contains a point and all the points within a certain distance from it. Open balls are often used to define neighborhoods in topology.

5. Can open balls be used in any metric space?

Yes, open balls can be used in any metric space. The definition of an open ball is based on the concept of distance in a metric space, so as long as a metric space is defined, open balls can be used.

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