Equivalence Relations on [0,1]x[0,1] and Hausdorff Spaces

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SUMMARY

The discussion centers on the equivalence relation defined on the set [0,1] × [0,1], where two points (x_0, y_0) and (x_1, y_1) are equivalent if x_0 = x_1 > 0. It concludes that the point (0,0) does not satisfy the properties of an equivalence relation, as it is not equivalent to itself. Furthermore, the conversation highlights that the singleton equivalence classes, such as {(0,y)}, are significant in understanding the topology of the space, particularly regarding the nature of open sets in the quotient topology.

PREREQUISITES
  • Understanding of equivalence relations in mathematics
  • Basic knowledge of topology, specifically Hausdorff spaces
  • Familiarity with quotient topology concepts
  • Experience with Cartesian products of sets
NEXT STEPS
  • Research the properties of Hausdorff spaces in topology
  • Study equivalence relations and their implications in set theory
  • Explore the concept of quotient topology and its applications
  • Examine examples of singleton equivalence classes in various topological spaces
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Mathematicians, particularly those specializing in topology, students studying advanced mathematics, and anyone interested in the properties of equivalence relations and Hausdorff spaces.

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We have a equivalence relation on [0,1] × [0,1] by letting (x_0, y_0) ~ (x_1, y_1) if and only if x_0 = x_1 > 0... then how do we show that X\ ~is not a Hausdorff space ?
 
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Wait, so (0,0) is not equivalent to itself? Then it's not an equivalence relation?
 
g_edgar said:
Wait, so (0,0) is not equivalent to itself? Then it's not an equivalence relation?

I think topologists usually don't bother explicitly describing eqivalence classes with a single member, so in this case each {(0,y)} would be a singleton equivalence class. (What can you say about open sets around {(0,y)} and {(0,y')} in the quotient topology, where y and y' are distinct?)
 
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