Is the empty set a compatible subset of a dense set?

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Discussion Overview

The discussion revolves around the concept of density in sets, particularly focusing on whether the empty set can be considered a compatible subset of a dense set. Participants explore the implications of density in both ordered and unordered sets, as well as the relationship between subsets and their density properties.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that if A is a dense set, then all subsets of A are dense, raising questions about the compatibility of the empty set as a subset.
  • Others challenge the assertion that all subsets of a dense set are dense, providing counterexamples such as the set of rational numbers and its subsets.
  • A participant argues that in point set topology, all subsets of a point set must be dense, questioning how to isolate points within such a set.
  • There are claims that non-empty subsets of dense sets can exist that are not dense, citing examples from the real numbers and rational numbers.
  • Some participants express confusion regarding the definition of "close" in the context of dense sets and how it relates to the concept of density.
  • Clarifications are made regarding the terminology used, particularly the distinction between the adjective "dense" as it applies to sets versus its relational context in topology.
  • A suggestion is made that the set of complex numbers may satisfy certain conditions related to density and ordering.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as multiple competing views remain regarding the properties of dense sets and their subsets. The discussion includes both agreement on certain definitions and disagreement on the implications of those definitions.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about ordered versus unordered sets, as well as the definitions of density and closeness in various contexts. Some participants express uncertainty about the terminology and its implications in set theory.

  • #31
SW VandeCarr said:
Simply that if every subset of T (as 'morphism' says in post 24) is dense in itself, then the empty set must be dense in itself as a subset of T. Since the empty set contains no points of the point set T, this would appear to me to be a contradiction.

quadraphonics said:
No, the usual definition of a dense set is: "If set X is dense in set Y, any point in set Y can be 'well-approximated' by a point in set X". If set Y is the empty set, that is trivially true, since there are no points in Y to worry about.
Or, a little more generally, a set X is dense in set Y if and only if Y is contained in the closure of Y. As has been said repeatedly, every set, no matter what it is a subset of, is dense in itself because every set is contained in its own closure. The empty set is itself a closed set: The closure of the empty set is itself which certainly contains itself. That has nothing at all to do with whether it contains any points.
 
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  • #32
HallsofIvy said:
Or, a little more generally, a set X is dense in set Y if and only if Y is contained in the closure of Y. As has been said repeatedly, every set, no matter what it is a subset of, is dense in itself because every set is contained in its own closure. The empty set is itself a closed set: The closure of the empty set is itself which certainly contains itself. That has nothing at all to do with whether it contains any points.

Thank you Quadaphonics and HallsofIvy. The key, at least to me, is that the empty set is in fact considered 'dense in itself.'
 
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  • #33
Since the definition of "closure" of a set is the set itself union all limit points, every set is contained in its own closure: every set is dense in itself. I don't see how that is "key" to anything!

The whole point of "density", typically, is to have some comparatively small set dense in a large set (as the countable rationals in the uncountable reals).
 

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