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

[kntsy]

1. Is closed ball the derived set of open ball?
2. In discrete metric space, boundary of a set is always the empty set?

 

[losiu99]

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?

 

[Landau]

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.

 

[kntsy]

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.

 

[Landau]

But still:

The least you can do is write out the definitions and give your reasoning.

 

[kntsy]

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?

\partial S=\overline{S} \bigcap \overline{X-S}

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.

 

[kntsy]

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

 

[HallsofIvy]

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.

 

[jessicaw]

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.