Hausdorff Space and finite complement topology

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SUMMARY

The finite complement topology on the real numbers R is not Hausdorff, as it fails to provide disjoint neighborhoods for every pair of distinct points. In this topology, open sets are defined as unions of intervals that exclude only finitely many points. For example, the open set (-inf, 1) U (1, inf) does not allow for the existence of disjoint neighborhoods for points such as 0 and 2, as their neighborhoods must be rays rather than traditional open intervals. This conclusion highlights a fundamental property of the finite complement topology that distinguishes it from Hausdorff spaces.

PREREQUISITES
  • Understanding of topology concepts, specifically open sets and neighborhoods.
  • Familiarity with the definition of Hausdorff spaces.
  • Knowledge of finite complement topology and its properties.
  • Basic comprehension of real number intervals and their unions.
NEXT STEPS
  • Study the properties of Hausdorff spaces in more detail.
  • Explore examples of other topologies, such as discrete and indiscrete topologies.
  • Learn about the implications of non-Hausdorff spaces in mathematical analysis.
  • Investigate the role of open sets in various topological spaces.
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Mathematicians, students of topology, and anyone interested in understanding the properties of different topological spaces and their implications in analysis.

Pippi
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I want to come up with examples that finite complement topology of the reals R is not Hausdorff, because by definition, for each pair x1, x2 in R, x1 and x2 have some disjoint neighborhoods.

My thinking is as follows: finite complement topology of the reals R is a set that contains open sets of the form, (- inf, a1) U (a1, a2) U ... U (an, inf). The complement of an open set is the finite elements {a1, a2, ... an}. However, any two points I pick out of any open set have disjoint neighborhoods. How is it possible?
 
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OK, pick two elements, how do they have disjoint neigborhoods?
 
Ok, I see where my reasoning was wrong. The neighborhoods have to be open sets in this topology. If I pick two points, {0, 2}, in the open set (-inf, 1) U (1 inf), their neighborhoods can't be any open intervals and must be rays.

Cheers!
 

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