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

In summary: 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.
  • #1
kntsy
82
0
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
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?
 
  • #3
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.
 
  • #4
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.
 
  • #5
But still:
The least you can do is write out the definitions and give your reasoning.
 
  • #6
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:
  • #7
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
 
  • #8
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.
 
  • #9
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.
 

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

1. What is basic topology?

Basic topology is a branch of mathematics that deals with the study of the properties of geometric objects and their relationships, without the use of measurements or numbers. It aims to understand the underlying structure and patterns of spaces and shapes.

2. Why are some questions in basic topology considered "stupid"?

The term "stupid" is often used to refer to questions that may seem obvious or trivial to those who are well-versed in the subject. However, in reality, there are no stupid questions in basic topology as it is a complex and abstract field that can be difficult for beginners to grasp.

3. Is basic topology useful in real life?

Although basic topology may not have direct applications in everyday life, it has many practical uses in fields such as physics, engineering, computer science, and economics. It provides a powerful framework for understanding and solving problems involving spatial relationships and structures.

4. What are some common concepts in basic topology?

Some common concepts in basic topology include open and closed sets, continuity, compactness, connectedness, and topological spaces. These concepts help define and characterize the properties of spaces and shapes in a topological sense.

5. How can one improve their understanding of basic topology?

To improve understanding of basic topology, it is important to have a strong foundation in mathematical concepts such as set theory, logic, and algebra. One can also benefit from practicing with examples and solving problems, as well as seeking guidance from books, lectures, and online resources.

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