Does Every Open Set Contain a Compact Set?

In summary, it is true that every open set contains a compact set. This is demonstrated by the fact that even the empty set, which is open, contains a compact set. However, it should be noted that this does not prove what the original poster was asking for, as the empty set is not a valid example. Additionally, it is worth mentioning that every open set also contains the empty open set, which contains the empty compact set.
  • #1
AxiomOfChoice
533
1
Is it true that every open set contains a compact set?
 
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  • #2
Well, the empty set is open. And yes, even the empty set contains a compact set.
 
  • #3
Showing that one open set contains a compact set does not prove what the OP wanted!

However, I can finish by showing the "easy" part: If A is not empty, then it contains, say, "a". The singleton set "a", since it is finite, is compact.
 
  • #4
My point was, that every open set contains the empty open set, which contains the empty compact set. So I was not really doing *one* open set after all.
 

1. What is an open set?

An open set is a set that contains all of its limit points. This means that any point in the set has a neighborhood that is also contained within the set. In other words, there are no boundary points in an open set.

2. How is an open set different from a closed set?

An open set does not include its boundary points, while a closed set includes all of its boundary points. Another way to think about it is that an open set is "open" because it does not contain its endpoints, while a closed set is "closed" because it contains all of its endpoints.

3. What is the definition of a compact set?

A compact set is a set that is both closed and bounded. This means that every sequence in the set has a convergent subsequence, and the set is contained within a finite interval or region. In other words, a compact set is a set that is "small" in some sense.

4. Can a set be both open and compact?

No, a set cannot be both open and compact. This is because an open set does not contain its boundary points, while a compact set includes all of its boundary points. Therefore, an open set cannot be bounded, and a compact set cannot be unbounded.

5. What is the relationship between open sets and continuous functions?

In general, continuous functions map open sets to open sets. This means that if a function is continuous, then the preimage of an open set will also be open. However, this is not always true, as there are some special cases where a continuous function can map open sets to closed sets or vice versa.

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