- #31
Andres316
- 8
- 1
Ok, thank you
Quotient maps, group action, open maps
Click For Summary
Discussion Overview
The discussion revolves around the properties of quotient maps in the context of a topological group acting continuously on a topological space. Participants explore whether the quotient map from a space X to its orbit space X/G is open, examining the implications of group actions and continuity in topology.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants propose that if G acts continuously on X, then for any open set U in X, the preimage under the action map is open in G x X.
- Others argue that the openness of the set g.U, defined as {g.x | x ∈ U}, should follow from the continuity of the group action.
- A participant suggests that the map defined by left multiplication by g is continuous and has a continuous inverse, indicating that it is a homeomorphism.
- Some express caution regarding the distinction between open and continuous properties in topology, emphasizing the need for rigorous verification of each step in the argument.
- There is discussion about the necessity of the group action being bijective and whether the conditions of the problem guarantee this property.
- One participant raises a concern about the implications of the group action being trivial and its effect on the openness of g.U.
- Another participant questions whether a group action is necessarily an element of GL(X) and discusses the relationship between topological groups and Lie groups.
Areas of Agreement / Disagreement
Participants express a mix of agreement and uncertainty regarding the implications of the group action on the openness of the quotient map. While some points are clarified, there remains no consensus on several aspects, including the necessity of bijectiveness and the nature of the group action.
Contextual Notes
Participants note that the continuity of group operations and inverses is essential, but there are unresolved questions about the specific conditions required for the action to be classified as a homeomorphism. Additionally, the discussion highlights the potential for confusion regarding the definitions and properties of topological groups.
Similar threads
Undergrad
Homemorphism in quotient topology
- · Replies 5 ·
- · Replies 15 ·
- · Replies 1 ·
- · Replies 8 ·
- · Replies 6 ·
- · Replies 2 ·
Undergrad
Does L_g pass to the quotient G/H?
- · Replies 4 ·
- · Replies 8 ·
- · Replies 43 ·
Graduate
Question about lifting of a function
- · Replies 2 ·