Interior points proof where one set is a subset of the other

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Discussion Overview

The discussion revolves around proving the relationship between the interior points of two sets, specifically that if set A is a subset of set B, then the interior points of A are also a subset of the interior points of B. The scope includes theoretical reasoning and mathematical proof.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks how to prove that the interior points of A are a subset of the interior points of B, seeking guidance on the approach.
  • Another participant requests clarification on the definition of an interior point and inquires about the original poster's (OP) attempts and current challenges.
  • A participant proposes that if a is an interior point of A, then there exists an open set Q in A containing a. They argue that since A is a subset of B, both a and Q must also be in B, leading to the conclusion that a is an interior point of B.
  • A later reply reiterates the previous point about the relationship between interior points of A and B, while expressing frustration that the OP was not allowed to discover this on their own.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the approach to the proof, and there is a mix of support and critique regarding how the discussion is being conducted.

Contextual Notes

There is a lack of clarity regarding the definitions being used, particularly what constitutes an interior point, and the discussion does not resolve the mathematical steps involved in the proof.

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How would I go about proving that if A is a subset of B then the interior points of A are a subset of the interior points of B?
 
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What did you try?? Where are you stuck? What is your definition of interior point?
 
If a is an interior point of A, the there is an open set Q in A containing a. Since A is a subset of B, a and Q are in B. Therefore a is an interior point of B.
 
mathman said:
If a is an interior point of A, the there is an open set Q in A containing a. Since A is a subset of B, a and Q are in B. Therefore a is an interior point of B.

It would have been so much better if you let the OP find this himself.
 

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