Topology Question (Normal Spaces)

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SUMMARY

The discussion centers on the properties of a specific topology on the Real line, which lacks disjoint open sets and consequently has no disjoint closed sets. The user questions whether a topology without disjoint closed sets can still be classified as normal. They reference Munkres' definition of normality, which states that single-point sets must be closed. Additionally, the user has demonstrated that the topology is not Hausdorff and seeks clarification on the implications of this result regarding normality.

PREREQUISITES
  • Understanding of topology concepts, specifically normal and Hausdorff spaces.
  • Familiarity with Munkres' "Topology" textbook and its definitions.
  • Knowledge of open and closed sets in topological spaces.
  • Basic principles of set theory as they apply to topology.
NEXT STEPS
  • Study the definitions and properties of normal spaces in topology.
  • Explore the implications of a topology being Hausdorff and its relationship to normality.
  • Review examples of non-normal topologies to understand their characteristics.
  • Investigate the concept of closed sets and their role in defining topological properties.
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Mathematicians, students of topology, and anyone interested in the properties of topological spaces, particularly those studying normal and Hausdorff spaces.

BSMSMSTMSPHD
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Not sure where to put a question about topology, but I'll try here.

I'm trying to show that a certain topology for the Real line is not normal. The topology in question has no disjoint open sets (they are all nested) and therefore, no disjoint closed sets.

If a topology has no disjoint closed sets, can it still be normal? I'd say not, but the only definition I have for normal requires me to start with disjoint closed sets, which in this case is impossible.

Another possible approach... I've been able to show that the topology in question is not Hausdorff. Does that necessarily mean it is not normal?

Any help is appreciated.
 
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According to munkres, for normality 1 point sets must be closed.
 

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