Add 2 Dense Sets for Non-Dense Set Result

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SUMMARY

The discussion centers around the properties of dense sets in mathematical contexts, specifically addressing whether the union of two dense sets can result in a non-dense set. Participants clarify that if two sets, X and Y, are both dense in a set Z, their union will also be dense in Z. The conversation also touches on the definitions of uncountability and density, emphasizing that uncountable sets cannot be put into a one-to-one correspondence with natural numbers, and that density is related to topology rather than countability.

PREREQUISITES
  • Understanding of set theory concepts, particularly dense and uncountable sets.
  • Familiarity with topology and metric spaces.
  • Knowledge of cardinality and its implications in mathematics.
  • Basic understanding of limits and convergence in mathematical analysis.
NEXT STEPS
  • Study the properties of dense sets in topology, focusing on definitions and examples.
  • Explore Cantor's diagonal argument and its implications for different sizes of infinity.
  • Learn about metric spaces and their role in defining density.
  • Investigate the concept of cardinality and how it applies to various sets in mathematics.
USEFUL FOR

Mathematics students, educators, and anyone interested in advanced set theory, topology, and the concept of infinity.

  • #31
cragar said:
Ok I see , I am very much enjoying this conversation .

I'm glad you find this forum informative! :biggrin:
 
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  • #32
You may want to read "The pea and the sun" by Wapner. It has some very informative things on infinity and it's paradoxes...
 
  • #33
thanks for the recommendation
 

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