Are Closed and Open Balls in Topology as Simple as They Seem?

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Discussion Overview

The discussion revolves around the properties of closed and open balls in topology, particularly in the context of discrete metric spaces. Participants explore definitions, boundaries, and the relationships between these concepts, raising questions about their implications and interpretations.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • Some participants question whether a closed ball is the derived set of an open ball.
  • There is a query about the definition of a boundary in terms of closures of sets, with a specific interest in properties within discrete metric spaces.
  • One participant asserts that in a discrete metric space, the boundary of a set is always the empty set, while another counters that the boundary is the set itself and that the interior is empty.
  • Some participants express frustration about the expectation for others to provide definitions and reasoning, suggesting a homework-related context.
  • A participant notes that in discrete metric spaces, the derived set of an open ball is smaller than the closed ball.
  • There is confusion expressed regarding the nature of boundaries and interiors in discrete topology, with differing interpretations presented.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the properties of boundaries and interiors in discrete metric spaces, with competing views on whether the boundary is empty or equivalent to the set itself. The discussion remains unresolved with multiple interpretations presented.

Contextual Notes

Participants reference definitions and properties that may depend on specific interpretations of topology and metric spaces, indicating potential limitations in understanding or assumptions made in the discussion.

kntsy
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1. Is closed ball the derived set of open ball?
2. In discrete metric space, boundary of a set is always the empty set?
 
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2. What is the definition of a boundary in terms of closures of sets? What is the nice property of closure in discrete metric space?
 
The least you can do is write out the definitions and give your reasoning. Now it sounds like you want us to do your homework.
 
Landau said:
The least you can do is write out the definitions and give your reasoning. Now it sounds like you want us to do your homework.

no, they are inspired from the HW solutions and i just cannot understand them.
 
But still:
The least you can do is write out the definitions and give your reasoning.
 
losiu99 said:
2. What is the definition of a boundary in terms of closures of sets? What is the nice property of closure in discrete metric space?

[tex]\partial S=\overline{S} \bigcap \overline{X-S}[/tex]

I think the nice property if closure of a set is the set itself?

so i dedece from the equation that the boundary is empty set.

Oh thanks.
 
Last edited:
Landau said:
But still:

Yes i know the definition and realize that in discrete metric derived set of open ball is smaller than the closed ball. Thanks
 
kntsy said:
2. In discrete metric space, boundary of a set is always the empty set?
On the contray. In a discrete topology, the boundary of a set is the set itself. It is the interior of the set that is empty.
 
HallsofIvy said:
On the contray. In a discrete topology, the boundary of a set is the set itself. It is the interior of the set that is empty.

Quite confused, i think in discrete topology, the boundary is empty because for very small ball only contains the element itself and no others. Therefore there is no boundary points.
Also, the interior point is the element itself as the open ball contains only the element.
 

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