Discussion Overview
The discussion revolves around the properties of open sets and their closures in the context of topology. Participants explore whether the closure of an open set A, which is not closed, is strictly larger than A itself, and examine related concepts such as connectedness.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants propose that if A is an open set and not closed, then its closure Cl(A) must contain at least one point not in A.
- Others argue that if the whole space is considered, it is closed, and thus Cl(A) cannot be larger than the whole space.
- A participant notes that if Cl(A) is not strictly bigger than A, it must be exactly A, which contradicts the assumption that A is not closed.
- Another participant introduces the concept of connectedness, stating that in a connected space, the closure of a non-empty proper open subset A will be strictly larger than A.
Areas of Agreement / Disagreement
There is no consensus on whether the closure of an open set A is strictly larger than A, as participants present differing viewpoints and conditions regarding the nature of the space and the properties of the sets involved.
Contextual Notes
Participants reference the axioms of topology and the definitions of open and closed sets, but there are unresolved assumptions regarding the nature of the space and the specific properties of the sets being discussed.