Discussion Overview
The discussion revolves around the question of whether there are "nice" proofs, particularly at an undergraduate level, that demonstrate the connectedness of the space ## X:=R^2 - \{p,q\} ## when two points are removed. The focus is on exploring various approaches to establish this property without relying on more advanced results.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant suggests considering path-connectedness to show that ## X ## is connected, seeking a proof that avoids the need to establish that path-connectedness implies connectedness.
- Another participant questions whether proving that path-connectedness implies connectedness is indeed difficult, suggesting it is a standard undergraduate result.
- A participant proposes a method involving the contradiction that arises from assuming ## X ## is not connected, using a continuous mapping from an interval to demonstrate the connectedness of the interval itself.
- There is a discussion about proving the connectedness of the unit interval and how that leads to proving the connectedness of other sets, including the plane.
- One participant mentions the idea of using star-shaped sets to show that the twice punctured plane is connected, but expresses uncertainty about how to prove that.
- Another participant notes that a set is connected if every continuous map from it to a two-point set is constant, but acknowledges the difficulty in applying this to specific sets like intervals.
Areas of Agreement / Disagreement
Participants express differing views on the difficulty of proving that path-connectedness implies connectedness. There is no consensus on the best approach to prove the connectedness of ## X ##, and multiple methods are discussed without resolution.
Contextual Notes
The discussion highlights the reliance on various properties of connectedness and path-connectedness, as well as the challenges in establishing these properties for specific sets. Some assumptions and definitions are not fully explored, leaving certain aspects unresolved.