Constructing a Set with the following characteristics

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SUMMARY

The discussion revolves around constructing a set ScR where the set S, its interior, closure, closure of the interior, and interior of the closure are all distinct. The participant struggles with the concept, noting that if S is open, its interior equals S, while if S is closed, its closure equals S. The rational numbers (S = Q) are considered, but they fail to meet the criteria as their interior is empty and closure equals the real number line, leading to overlaps. The challenge lies in finding a set that satisfies all distinct conditions.

PREREQUISITES
  • Understanding of set theory concepts such as closure and interior of a set.
  • Familiarity with open and closed sets in topology.
  • Knowledge of rational and real number properties.
  • Basic comprehension of neighborhood definitions in mathematical analysis.
NEXT STEPS
  • Research the properties of open and closed sets in topology.
  • Explore examples of sets that demonstrate distinct interior and closure properties.
  • Learn about the concept of boundary points in relation to set closure.
  • Investigate advanced set constructions in topology, such as the Cantor set.
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Mathematics students, particularly those studying topology and set theory, as well as educators seeking to understand complex set relationships.

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Homework Statement



Construct a set ScR such that S, the interior of S, the closure of S, the closure of the interior of S, and the interior of the closure of S are all distinct (ie no 2 of them are equal)


Homework Equations


closure of S - smallest closed set containing S
interior of S - set of all points in S for which S is a neighbourhood



The Attempt at a Solution



I am really having trouble starting this...or more precisely, I'm having trouble seeing how this question is possible at all...How can a set, its interior, and its closure all be distinct? If a set is open, then its interior is simply equal to the set itself, so that leaves me with only closed sets to consider. But, if a set is closed, then it contains all of its boundary points, so the closure of S is equal to S...
so the only exception i could think of is the open/closed sets...

but so far the closest I've gotten to something that satisfies all of those things is the set of rational numbers
S = Q
int S = empty set
Closure of S = Real number line
but then closure of the interior of S = closure of the empty set = empty set
so that's equal to int S and doesn't work
and the interior of the closure of S = real number line = closure of S so that doesn't work either...help!
 
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I just finished answering this question from someone else. You can find the thread here:
<https://www.physicsforums.com/showthread.php?t=314690>
Enjoy!
 
Oh wow...awesome! thanks :) (its probably a classmate of mine haha...since both of us thought of rational numbers, and that's something we discussed in class)
 

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