Why Should We Consider Compactness in Sets?

  • Thread starter remaths
  • Start date
  • Tags
    Important
In summary, compactness is an important concept in mathematics because it shares many properties with finite sets, such as being closed, bounded, and having maxima or minima. This allows for easier proofs and analysis.
  • #1
remaths
2
0
Hi everyone,

I am asking very basic question. But I think there is some important concept I am missing.

why is compactness important or why should one look out for compact sets?

I understand that they are bounded and closed and functions on them have good properties wrt to continuity and maxima or minima.


thanks
remaths
 
Physics news on Phys.org
  • #2
I've said this before but it bears repeating: compactness is "the next best thing to finite"! Consider all of the properties of a finite set: It is closed; it is bounded; in the case of a metric space, there exist two points in the set having maximum distance between them; in the case of a finite set of real numbers it contains a smallest and a largest number. All of those are true for a compact set. In fact, in each case you can prove them by taking an open cover consisting of open sets about each point in the set and using compactness to reduce to a finite cover- and so to a finite set of points.
 
  • #3
Thank you for your reply
 
  • #4
HallsofIvy said:
compactness is "the next best thing to finite"

This is one of the most amazing one-line summaries of a mathematical concept I have ever seen. I was about to say this about 100 times less eloquently.
 
  • #5
halls beat me to it but i was going to parrot my analysis teacher and say exactly that
 

1. Why is compactness important in research studies?

Compactness is important in research studies because it allows for more efficient use of limited resources such as time, space, and materials. By keeping the research study compact, researchers can focus on specific variables and reduce the risk of confounding factors affecting their results. This also allows for more precise and accurate conclusions to be drawn from the study.

2. How does compactness impact the validity of a study?

Compactness can greatly impact the validity of a study. A compact study design can reduce the likelihood of errors and biases, as well as increase the internal validity of the study. This means that the results of the study are more likely to accurately reflect the relationship between variables being studied.

3. What are the benefits of compactness in experimental design?

The benefits of compactness in experimental design include increased control over variables, reduced risk of errors and biases, and improved precision and accuracy of results. Compactness also allows for more efficient use of resources, making the study more cost-effective.

4. Can a study be considered reliable if it is not compact?

While a study may still be reliable if it is not compact, there is a higher risk of errors and confounding factors affecting the results. Compactness is an important aspect of research design that can enhance the reliability of a study by minimizing these risks. However, other factors such as sample size and methodology also play a role in determining the reliability of a study.

5. How can compactness be achieved in a research study?

Compactness can be achieved in a research study by carefully selecting and controlling variables, limiting the scope of the study, and utilizing appropriate study designs such as randomized controlled trials. It is also important to consider the specific research question and objectives when designing a study to ensure that it remains compact and focused.

Similar threads

Replies
3
Views
781
Replies
11
Views
4K
Replies
4
Views
1K
  • Topology and Analysis
Replies
4
Views
2K
Replies
3
Views
2K
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
4
Views
705
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
Replies
4
Views
2K
Back
Top