How can you tell if a specific topological space is compact?

In summary, the conversation discusses the concept of compactness in topological spaces and various theorems that can help determine if a space is compact. It also clarifies the notion of a cover and the difference between a subset and a proper subset. The conversation ends with a discussion about the effectiveness of learning from the internet versus reading books.
  • #1
PBRMEASAP
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eddo's thread got me thinking: How can you tell if a specific topological space is compact? It seems like it would be hard to do just starting with the definition of compactness.
 
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  • #2
Yes, it would, that is why we have lots of theorems to help: the product of compact spaces is compact. Any closed subspace of a compact space is compact. Every compact metric space is sequentially compact. Usually, though I think it would be easier to show something wasn't compact by finding an explicit open subcover without refinement. Plus we often know a lot about the topological space other than simply that it is a topological space.
 
  • #3
Thanks, I'll look into those theorems you listed. Also, I wanted to make sure I have the notion of a cover straight. If a family of sets form a cover for a set A, then that means A is contained in the union of those sets, right? The reason I ask is that on the Wolfram Mathworld site, they say
A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose union is X.
In this case, we are talking about the whole topological space X, so is that why they say X is the union of a family of open sets? I just want to make sure that when talking about the compactness of a subset A of X, it is okay for A to be a proper subset of its cover, rather than equal to it.
 
  • #4
If A is a subset of a topological space, then it is almost certain that A must be a proper subset of the cover, otherwise A if it were equal to the union of the cover would be open, which is usually a way to be "not compact".
 
  • #5
Why don't people just go to the library and read books instead of trying to learn it from the internet? One cannot learn a subject by reading encyclopedia entries.
 
  • #6
matt:
Thanks for clearing that up!


mruncleramos said:
Why don't people just go to the library and read books instead of trying to learn it from the internet? One cannot learn a subject by reading encyclopedia entries.
Who is you? Where you comin' from? :cool:
 

1. What is compactness in topology?

Compactness in topology refers to a specific property of a topological space, which is a mathematical concept that describes the properties of a set of points and the relationships between them. A topological space is said to be compact if it has the property of being able to be fully covered by a finite number of open sets.

2. How can I tell if a topological space is compact?

There are several ways to determine if a topological space is compact. One way is to use the Heine-Borel Theorem, which states that a topological space is compact if and only if it is both closed and bounded. Another way is to use the definition of compactness, which states that every open cover of a compact space must have a finite subcover.

3. Can a topological space be both compact and non-compact?

No, a space cannot be both compact and non-compact. This is because the definition of compactness is a binary property, meaning that a space is either compact or it is not. However, it is possible for a space to be neither compact nor non-compact, in which case it is considered to be neither compact nor non-compact.

4. Are all metric spaces compact?

No, not all metric spaces are compact. A metric space is a specific type of topological space, and while some metric spaces may be compact, it is not a requirement for all metric spaces. Compactness is a property that is specific to a particular topological space, and cannot be generalized to all types of topological spaces.

5. What are the practical applications of understanding compactness in topology?

Understanding compactness in topology is important in various fields of mathematics, such as analysis, geometry, and differential equations. It also has practical applications in physics, engineering, and computer science, where it is used to study and solve problems related to continuity, convergence, and optimization.

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