Complete, Equivalent, Closed sets

Click For Summary

Discussion Overview

The discussion revolves around the properties of equivalent sets, specifically focusing on completeness and closedness. Participants explore whether these properties can be transferred between sets that are equivalent, as defined by cardinality and one-to-one correspondence.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether the completeness of set A implies completeness for set B when A and B are equivalent.
  • Others suggest that equivalence, defined by cardinality, does not guarantee that properties like closedness are preserved between sets.
  • A participant notes that while equivalent sets have the same cardinality, they may differ in closure properties, particularly in the context of open and closed intervals.
  • There is a proposal that if sets A and B are isometric, then the closedness of A might imply the closedness of B, although this is debated.
  • Another participant emphasizes that the only inherent measure of a set is its cardinality, which does not necessarily dictate closure properties.
  • A later reply introduces the concept of a metric preserving homeomorphism as a condition under which the properties may hold.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between equivalence and properties like completeness and closedness. There is no consensus on whether these properties can be transferred between equivalent sets.

Contextual Notes

Participants reference specific definitions of equivalence and cardinality, but there are unresolved assumptions regarding the implications of these definitions on set properties. The discussion also touches on the need for stricter conditions, such as isometry, to draw conclusions about closedness.

Somefantastik
Messages
226
Reaction score
0
If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete?

What if it is known that A is closed, can it then be said that B is also closed?
 
Physics news on Phys.org
Somefantastik said:
If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete?

What if it is known that A is closed, can it then be said that B is also closed?

Two sets are equal if they have the same elements. So the set A=B={[0,1]} is closed because it contains both 0 and 1. I don't know if you mean something different by using the term "equivalent".
 
Last edited:
Kolmogorov states that equivalents sets are those on which a one-to-one correspondence can be found.
 
Somefantastik said:
Kolmogorov states that equivalents sets are those on which a one-to-one correspondence can be found.

This means two sets have the same cardinality. Since both the open and closed sets on the real interval 0,1 have the same cardinality, A and B may nevertheless differ in terms of closure. If you wish to call the closed interval "complete" then the open interval would not be "complete".
 
Last edited:
SW VandeCarr said:
Since both the open and closed sets on the real interval 0,1 have the same cardinality

Can you clarify that statement a little?

Also, would making the requirement more strict, maybe saying if the sets A and B are isometric, then A closed implies B is closed?
 
Somefantastik said:
Can you clarify that statement a little?

If a < b, then |[a, b]| = |(a, b)| = |(a, b]| = |[a, b)| where |...| is the cardinality.
 
Somefantastik said:
Also, would making the requirement more strict, maybe saying if the sets A and B are isometric, then A closed implies B is closed?

What metric are you defining for your sets? The only inherent measure of a set is the number of elements it contains (ie its cardinality). The fact that sets A and B have the same cardinality doesn't imply that if A is closed, B must be closed.
 
Last edited:
Nevermind, I have the answer...it works when you have a metric preserving homeomorphism between the sets.
 

Similar threads

Undergrad Formalizing infinity sets
  • · Replies 3 ·
Replies
3
Views
391
High School Sets being equal and equivalent
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad How to find this equivalent of the material conditional?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate When can transitivity emerge from cycle-suppressing dynamics?
  • · Replies 1 ·
Replies
1
Views
482
Undergrad Set Theory - the equivalence relation on elements
  • · Replies 14 ·
Replies
14
Views
2K
Graduate Second order logic and completeness
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad How to determine if a set is a semiring or a ring?
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Geometric Point of View of sets
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
4K
Graduate On soundness and completeness of ZFC set theory
  • · Replies 6 ·
Replies
6
Views
2K