Correctness of Statements about Set A in Metric Space X

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Discussion Overview

The discussion revolves around the properties of the set $A=\{x\in X: \ 0\leq x\leq1\}$ within the metric space $X=\mathbb{R}\setminus \mathbb{Q}$. Participants explore various statements regarding the boundary, openness, compactness, and completeness of the set, engaging in a technical examination of definitions and implications in topology.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the boundary $\partial A$ is not a singleton, suggesting that statement (a) is incorrect.
  • There is a contention regarding the definition of the boundary, with some participants indicating that the complement $A^c$ was mistakenly referred to as the boundary.
  • Some argue that since the boundary is empty, $A$ could be both open and closed (clopen), challenging the correctness of statement (b).
  • Participants discuss the conditions for compactness, noting that $A$ is not closed, which leads to the conclusion that statement (d) is correct.
  • There is a proposal to check if $A$ is complete by examining Cauchy sequences, with some suggesting that a sequence of irrational numbers could converge to a rational limit.
  • Concerns are raised about whether certain sequences remain within the set $A$, particularly regarding the sequence $a_n=2^{\frac{1}{n}+1}$ and its elements in relation to $A$.
  • Another participant suggests using the sequence $2^{\frac{1}{n}-1}$ for $n\ge 2$, noting the need to exclude $n=1$ since it yields a rational number not in $A$.

Areas of Agreement / Disagreement

Participants express differing views on the properties of the set $A$, particularly regarding its boundary, openness, and completeness. No consensus is reached on the correctness of all statements, and multiple competing interpretations of the definitions are present.

Contextual Notes

Participants reference various definitions and properties from topology, indicating that the discussion is heavily dependent on these definitions and the interpretations of boundaries, compactness, and completeness. There are unresolved aspects concerning the application of these definitions to the specific set $A$.

  • #31
Furthermore, since $A$ has empty interior, $A$ cannot be open, making (c) the correct answer.
 
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  • #32
Euge said:
I do not see how $A$ has empty boundary. The closure of $A$ is $[0,1]$ since the irrationals are dense in the reals, and $A$ has empty interior since the rationals are dense in the reals. So wouldn't $\partial A$, being $\bar{A} - \operatorname{Int}(A)$, equal $[0,1]$?
The topology is $X=\mathbb R\setminus\mathbb Q$ with the usual metric.
So none of the rationals can be part of either the interior or the boundary.
So the closure of $A$ is $[0,1]\setminus \mathbb Q$, which is the same as the interior of $A$, making the boundary $\partial A$ empty, doesn't it?
 
  • #33
Looking back at the original question, it appears that I overlooked that $A$ inherits the subspace topology relative to $X$, not to $\Bbb R$. My apologies. Certainly $A = (0,1) \cap X$, which is relatively open in $X$, and $X - A = (\Bbb R - \Bbb [0,1]) \cap X$, which is relatively open in $X$; thus, $A$ is both open and closed in $X$, so that $A$ has empty boundary.
 

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