Hausdorff topology on five-element set that is not the discrete top.

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SUMMARY

The discussion centers on constructing a Hausdorff topology on the five-element set \{a,b,c,d,e\} that is not the discrete topology, as posed in the textbook "Introduction to Topology, Pure and Applied" by Adams and Franzosa. The participants conclude that it is impossible to create such a topology, as any finite Hausdorff space must be discrete. This is supported by the theorem stating that if a space is Hausdorff, then every single point subset is closed, leading to the conclusion that all four-element subsets must also be open, thus generating the entire power set.

PREREQUISITES
  • Understanding of basic topology concepts, including open and closed sets.
  • Familiarity with Hausdorff spaces and their definitions.
  • Knowledge of finite sets and their properties in topology.
  • Ability to interpret theorems related to topological spaces.
NEXT STEPS
  • Study the properties of Hausdorff spaces in more detail.
  • Explore the implications of the theorem stating that finite Hausdorff spaces are discrete.
  • Investigate examples of non-discrete topologies on infinite sets.
  • Learn about other types of topological spaces and their characteristics.
USEFUL FOR

Students of topology, mathematicians interested in set theory, and educators teaching concepts of topological spaces will benefit from this discussion.

ArcanaNoir
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Homework Statement



The textbook exercise asks for a Hausdorff topology on \{a,b,c,d,e\} which is not the discrete topology (the power set). It is from "Introduction to Topology, Pure and Applied", by Adams and Franzosa.

Homework Equations



Let X be a set.

Definition of topology (top for short): must include X and the empty set, must include all intersections of finitely many sets in the top, must include all unions of any sets in the top.

Definition of Hausdorff: for any two elements x and y of X, there must be disjoint open sets in the top such that one contains x and the other contains y.

The Attempt at a Solution



My thought is that this cannot be done. We have a theorem that if X is Hausdorff, then every single point subset of X is closed. This implies that all the four-element subsets of X are (in the topology) and open. But then the intersection of any of the four-element sets generates the rest of the powerset.
 
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That is correct. Any finite Hausdorff space must be discrete!

Your proof is correct as well!
 
Thanks micro :)
 

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