What is the Interior of the Closure of a Set in Rn?

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SUMMARY

The discussion centers on the relationship between the interior of the closure of a set S in Rn and the interior of S itself. It is established that the interior of the closure of S, denoted as int(closure(S)), is indeed a subset of the interior of S, int(S). The reasoning provided indicates that the interior of the closure can be viewed as the union of the interior of S and the interior of the boundary of S, which is empty. The example of the rationals in Rn illustrates the complexity of closures in different contexts.

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  • Knowledge of boundary points in topological spaces.
  • Basic grasp of rational numbers and their properties in Rn.
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wakko101
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Let S be a set in Rn, is it true that every interior point in the closure of S is in the interior of S? Justify.

ie. int(closure(S)) a subset of int(S)

It seems to me that it would be true...if you could say that the interior of the closure of S is the interior of S unioned with the interior of the boundary of S, then it would have to be true because the interior of S's boundary is the empty set.

Does that make sense?
 
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Nope. Let Q be the rationals. What the closure of Q^n? Etc.
 

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