Quotient maps, group action, open maps

  • Context: Graduate 
  • Thread starter Thread starter PsychonautQQ
  • Start date Start date
  • Tags Tags
    Group quotient
Click For Summary
SUMMARY

The discussion focuses on proving that the quotient map \( p: X \rightarrow X/G \) is open when a topological group \( G \) acts continuously on a topological space \( X \). The participants establish that the map \( m: G \times X \rightarrow X \) defined by \( (g,x) = gx \) is continuous, which implies that for any open set \( U \subseteq X \), the set \( g.U \) is also open. They conclude that the left multiplication map \( l_g: U \rightarrow gU \) is a homeomorphism, confirming that the quotient map retains the openness property.

PREREQUISITES
  • Understanding of topological groups and their properties
  • Familiarity with continuous functions and homeomorphisms
  • Knowledge of quotient spaces in topology
  • Basic concepts of group actions on topological spaces
NEXT STEPS
  • Study the properties of topological groups and their continuous actions
  • Explore the concept of quotient spaces in more depth
  • Learn about homeomorphisms and their significance in topology
  • Investigate examples of group actions on various topological spaces
USEFUL FOR

Mathematicians, particularly those specializing in topology, algebra, and group theory, as well as students seeking to deepen their understanding of quotient maps and group actions in topological contexts.

  • #31
Ok, thank you
 
  • Like
Likes   Reactions: WWGD
Physics news on Phys.org
  • #32
I think I saw this and someone thought it was homework. A bit ambitious homework if you ask me, but o.k., a mentor (not me) thought so. Try it again in the homework section or contact the mentor who classified it as homework.
 
  • Like
Likes   Reactions: Andres316

Similar threads

Undergrad Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
1K
High School Topological neigborhoods that are not intervals in the real line?
  • · Replies 15 ·
Replies
15
Views
3K
Graduate Regarding fibrations between smooth manifolds
  • · Replies 1 ·
Replies
1
Views
2K
High School Why Is the Trick for Open Sets in Quotient Topology Valid?
  • · Replies 8 ·
Replies
8
Views
3K
High School Is the Quotient Topology of Real Numbers Homeomorphic to Real Numbers?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Continuity of linear map from subspace of Euclidean space
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Does L_g pass to the quotient G/H?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Question about existence of path-lifting property
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Fiber bundle homeomorphism with the fiber
  • · Replies 43 ·
2
Replies
43
Views
6K
Graduate Question about lifting of a function
  • · Replies 2 ·
Replies
2
Views
2K