Can a Perfect Set Contain an Open Subset?

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Discussion Overview

The discussion revolves around the question of whether a perfect set in \(\mathbb{R}^k\) can contain an open subset. Participants explore examples and counterexamples related to this concept, touching on both theoretical and conceptual aspects.

Discussion Character

  • Debate/contested, Conceptual clarification

Main Points Raised

  • One participant proposes the question of whether an open set can exist within a perfect set E in \(\mathbb{R}^k\).
  • Another participant suggests that a closed interval is an example of a perfect set and illustrates this by mentioning the closed unit cube in \(\mathbb{R}^k\) and its open counterpart.
  • Some participants argue that while some perfect sets can contain open subsets, such as the example given, others, like the Cantor set, do not contain any open intervals.
  • A later reply acknowledges the complexity of the topic and expresses gratitude for the clarification provided.

Areas of Agreement / Disagreement

Participants express differing views on the existence of open subsets within perfect sets, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

The discussion highlights the need for clarity regarding definitions of perfect sets and open sets, as well as the implications of specific examples like the Cantor set.

kostas230
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Suppose we have a perfect set [itex]E\subset\mathbb{R}^k[/itex]. Is there an open set [itex]I\subset E[/itex]?
 
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I'm rusty at this. However, I understand that a closed interval is a perfect set. Take the closed unit cube in Rk, drop all boundary points leaving an open unit cube.
 
Sometime's it's true (like mathman's example), and sometimes it's false. For example the Cantor set is a perfect set but contains no open interval inside of it
 
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Yeah, I just found out that. Thank you :)
 

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