Question concerning compact subtopologies on Hausdorf spaces

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In summary, the conversation is about a topology exam and a question regarding a Hausdorf space (X,T) and its compact subsets. The goal is to prove the existence of a coarser topology on (X,T) where closed subsets also imply compactness. The suggestion of using the trivial topology is discussed.
  • #1
Kalimaa23
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Greetings,

I'm helping out a student with her upcoming topology exam and something has be stomped. It's probably simple but I'm not seeing it at the moment.

Consider a Hausdorf space (X,T). Any compact subset of X is therefore closed.

The question is to prove the existence of a coarser topology on (X,T) so that closed also implies compactness. I'm basically trying to find a coarser topology on X that makes it compact.

Thanks in advance.
 
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  • #2
What about the trivial topology (= only empty and X are open)?
 
  • #3
That had occurred to me, but I though there might be something you could prove about a topology finer than the trivial one. But I guess it fits the bill.
 

Related to Question concerning compact subtopologies on Hausdorf spaces

1. What is a compact subtopology?

A compact subtopology is a subset of a larger topological space that has the same open sets as the larger space, but is itself a topological space with a smaller set of points.

2. How is a compact subtopology different from the original topological space?

A compact subtopology is a subset of the original topological space that has the same open sets as the original space, but is itself a topological space with a smaller set of points. This means that the compact subtopology is a smaller, more condensed version of the original space.

3. How can compact subtopologies be applied in scientific research?

Compact subtopologies are often used in mathematics and physics research, particularly in the study of dynamical systems and chaos theory. They can also be used to analyze the behavior of complex systems and to study the properties of certain topological spaces.

4. Are there any limitations to compact subtopologies?

One limitation of compact subtopologies is that they can only be defined on Hausdorff spaces, which are topological spaces that satisfy a certain separation axiom. Additionally, not all topological spaces have a compact subtopology, so their use may be limited in certain cases.

5. Can compact subtopologies be used in real-world applications?

Yes, compact subtopologies have practical applications in fields such as engineering, computer science, and economics. They can be used to model and analyze complex systems, and to make predictions about their behavior. Compact subtopologies are also used in data compression algorithms and in the design of efficient networks and communication systems.

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