Do Sets in Discrete Topological Spaces Have Boundaries?

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SUMMARY

In discrete topological spaces, every subset has an empty boundary. This conclusion arises from the definition of the boundary of a set A, which is the intersection of the closure of A and the closure of its complement A^c. In discrete spaces, all subsets are both open and closed, leading to the closure of A being A and the closure of A^c being A^c, resulting in an empty intersection. This aligns with the intuitive understanding of discrete spaces, where all points are isolated.

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  • Understanding of discrete topological spaces
  • Familiarity with the concepts of closure and boundary in topology
  • Knowledge of open and closed sets
  • Basic understanding of set theory
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Mathematicians, students of topology, and anyone interested in the properties of discrete topological spaces and their boundaries.

Riemannliness
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Do sets in a discrete topological space have boundaries?
 
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Every subset of a discrete topological space has empty boundary. To see this recall the definition of the boundary of a set A as the intersection of the closure of A, and the closure of the complement of A, in a discrete space all subsets are both open and closed, so the closure of A is simply A and the closure of A^c is A^c, their intersection is empty. This should also fit our intuition of what discrete spaces are, spaces where all points are isolated, such as Z as a subspace of R (in the standard topology).
 
An addition to what Mandark said, it can be shown that a set has an empty boundary if and only if it is both open and closed.
 

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