The discussion revolves around the properties of compact subsets within a metric space, specifically examining the relationship between a set B, its interior Int(B), and its boundary bd(B). Participants are questioning the validity of the equality B = Int(B) ∪ bd(B).
The discussion is ongoing, with participants raising questions about the definitions and properties of boundary points in relation to closed sets. There is an exploration of the implications of compactness in metric spaces, but no consensus has been reached.
Participants note that in metric spaces, compact sets are closed, which may influence the understanding of boundary points. The discussion also touches on the nature of Hausdorff spaces, which is relevant to the properties being examined.
HallsofIvy said:In a metric space every compact set is closed (actually in every topological space).