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

In summary, the conversation discusses the textbook exercise of finding a Hausdorff topology on a set that is not the discrete topology. The concept of topology and Hausdorff is defined, and it is concluded that it cannot be done as any finite Hausdorff space must be discrete. A proof is also acknowledged and confirmed to be correct.
  • #1
ArcanaNoir
779
4

Homework Statement



The textbook exercise asks for a Hausdorff topology on [itex] \{a,b,c,d,e\}[/itex] 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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  • #2
That is correct. Any finite Hausdorff space must be discrete!

Your proof is correct as well!
 
  • #3
Thanks micro :)
 

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

What is Hausdorff topology on a five-element set?

Hausdorff topology is a mathematical concept that describes a way of organizing the elements of a set in a way that allows for the definition of neighborhoods and limits. In particular, Hausdorff topology on a five-element set is a way of defining the relationships between the elements of a five-element set in a way that satisfies certain properties.

What are the properties of Hausdorff topology on a five-element set?

There are several properties that define Hausdorff topology on a five-element set. These include the existence of open sets that contain each element and the ability to define neighborhoods for each element. Additionally, Hausdorff topology on a five-element set satisfies the Hausdorff separation axiom, which states that any two distinct elements of the set have disjoint neighborhoods.

How does Hausdorff topology on a five-element set differ from discrete topology?

Hausdorff topology and discrete topology are two different ways of organizing the elements of a set. While discrete topology defines each element as an open set, Hausdorff topology only requires the existence of open sets containing each element. In other words, Hausdorff topology allows for the potential overlap of open sets, while discrete topology does not.

Can Hausdorff topology on a five-element set be visualized?

Yes, it is possible to visualize Hausdorff topology on a five-element set. One way to do so is by representing the five elements as points on a graph, with lines connecting elements that have open sets containing both of them. This visual representation can help to understand the relationships between the elements and the properties of Hausdorff topology.

What are some real-world applications of Hausdorff topology on a five-element set?

Hausdorff topology on a five-element set has applications in various fields, including computer science, physics, and economics. For example, it can be used to model the relationships between different variables in a system, such as the interactions between particles in a physical system or the connections between different market factors in economics.

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