Discussion Overview
The discussion revolves around finding an example of a set X with two topologies T and S such that the identity function from (X,T) to (X,S) is continuous but not homeomorphic. The scope includes conceptual understanding of topological properties and definitions related to continuity and homeomorphism.
Discussion Character
- Homework-related
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant requests an example of a set X with two topologies T and S where the identity function is continuous but not homeomorphic, expressing feelings of being overwhelmed by the generality of the problem.
- Another participant asks for definitions of "continuous" and "homeomorphic," suggesting the post belongs in the Homework Help section.
- A participant provides standard definitions for continuity and homeomorphism, indicating that the identity function must fail at least one property of homeomorphism to not be homeomorphic.
- One participant suggests that the identity function must fail to meet one of the four properties of homeomorphism while remaining continuous, prompting further exploration of which property that might be.
- Another participant asserts that they are looking for a set X where the two topologies are never homeomorphic, indicating a belief that this is a general property of topological spaces.
- A suggestion is made to consider a two-point set, implying that this could serve as a suitable example.
- A participant inquires about the use of a particular point topology, seeking clarification on the approach being suggested.
- A later reply indicates that the participant has resolved their question independently, expressing gratitude for the help received.
Areas of Agreement / Disagreement
Participants express varying levels of understanding and approaches to the problem, with some disagreement on the complexity of the task and the definitions involved. The discussion does not reach a consensus on a specific example.
Contextual Notes
There are unresolved aspects regarding the specific properties of the topologies T and S that would lead to the identity function being continuous but not homeomorphic. The discussion also reflects varying interpretations of the definitions provided.