Discussion Overview
The discussion revolves around the concept of limit points in the context of metric spaces, specifically questioning whether a limit point can exist outside the set E. Participants also explore the definition of perfect sets and provide examples of closed sets that are not perfect.
Discussion Character
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants define a limit point as a point p where every neighborhood contains a point q≠p in the set E.
- There is a question posed about whether there exist limit points that are not included in the set E, specifically using the open interval E = (0,1) as an example.
- Participants are prompted to think of real numbers outside of (0,1) that could be limit points of this set.
- One participant asks for an example of a perfect set, defined as a closed set where every point is a limit point.
- Another participant suggests that [0, 1] could serve as an example of a closed set, while questioning the existence of closed sets that are not perfect.
- It is noted that adding isolated points, such as {0}, to the usual topology could lead to closed sets that are not perfect, with (0, 1] being mentioned as a candidate.
- A later reply clarifies that (0, 1] is not closed and provides examples of closed non-perfect sets, such as {2} or the union of [0,1] with {2}.
Areas of Agreement / Disagreement
The discussion contains multiple competing views regarding the existence of limit points outside of the set E and the characteristics of perfect sets. There is no consensus on the examples provided or the definitions applied.
Contextual Notes
Participants express uncertainty about the definitions and properties of limit points and perfect sets, and there are unresolved questions regarding the nature of closed sets in relation to perfection.