The discussion revolves around the concept of closed sets in the context of set theory, specifically focusing on the proof that every closed set contains all its accumulation points. The original poster expresses uncertainty about how to approach this proof and seeks assistance.
The conversation is ongoing, with participants highlighting the need for clear definitions before proceeding with any proofs. The original poster has provided an example of a closed interval but acknowledges a lack of formal definitions from their studies. This indicates a productive direction towards clarifying foundational concepts.
The original poster mentions that they have not yet covered the definitions of closed and open sets in their studies, which may limit their ability to engage fully with the proof discussion.
HallsofIvy said:What definition of "closed set" are you using? There are several different but equivalent definitions of "closed set"- and in mathematics, proofs often use the specific words of definitions. No one can help you prove this without knowing what definition you are using.
HallsofIvy said:If you do not know the definition of "closed set", then you cannot possibly prove anything about them! Examples will not provide proofs. Since you have a specific definition of "closed interval" you might be able to prove that a closed interval contains all its accumulation points. What is your definition of "accumulation point"?