Is the Interior of A Equal to the Interior of the Closure of A?

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Discussion Overview

The discussion revolves around the question of whether the interior of a set A is equal to the interior of the closure of A within a given topological space X. The context includes theoretical exploration of topology, specifically in relation to the set of rational numbers within the real numbers.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions if the interior of A is equal to the interior of the closure of A, suggesting this seems reasonable based on a visual interpretation of A as a disc.
  • Another participant provides a specific example using the set of rational numbers between 0 and 1, prompting further analysis of the interior and closure of this set.
  • A participant explains that the interior of the set of rational numbers is the empty set, while the closure is the interval [0, 1], leading to the conclusion that the interior of the closure is (0, 1).
  • A later reply confirms the understanding of the properties of the sets discussed, indicating agreement with the previous analysis.

Areas of Agreement / Disagreement

While there is some agreement on the properties of the sets involved, the initial question regarding the generality of the statement remains unresolved, as the discussion does not reach a consensus on whether the interior of A is always equal to the interior of the closure of A.

Contextual Notes

The discussion is limited to the specific example of rational numbers in the real numbers and does not explore other potential topological spaces or sets that may provide counterexamples or support for the initial question.

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If A \subset X where X has a topology, is it generally true that the interior of A is equal to the interior of the closure of A? This seems very reasonable to me, but probably only because I'm visualizing A as a disc in the real plane. If it isn't true, what would be a counterexample?

thanks
 
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Take X to be the set of real numbers with the usual topology.

Let A be the set of all rational numbers between 0 and 1 (inclusive).

What is the interior of A? What is the closure of A?

What is the interior of the closure of A?
 
HallsofIvy said:
Take X to be the set of real numbers with the usual topology.

Let A be the set of all rational numbers between 0 and 1 (inclusive).

What is the interior of A? What is the closure of A?

What is the interior of the closure of A?

Okay this is a good example! Let me see if I have it straight. The interior of A is the largest open set contained in A, which would be the empty set in this case. I say this because the set has no interior points, or points for which a neighborhood can be found containing only points in A. The closure of A is the smallest closed set containing A, which is the interval [0, 1]. I say this because for every point in [0, 1], every neighborhood of that point contains points in A. But the interior of [0, 1] is (0, 1), which is not the empty set.

Did I understand correctly? I am a little foggy on the properties of real numbers, so I can't really back up my claims about the density of rationals at the moment.

Thanks for your help!
 
yes...you've got it right...
 

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