Advanced Compactness and Open Cover Applications

  • 11th Grade
Reviewed by Cierra Henderson
Cierra Henderson, MBA |
K-12 Expert
Review Board Member
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
 , MBA
By Thames
T
Thames
Community Contributor
Quizzes Created: 8156 | Total Attempts: 9,588,805
| Questions: 14 | Updated: Jan 21, 2026
Please wait...
Question 1 / 15
🏆 Rank #-- ▾
Score 0/100

1) Every closed interval in ℝ is compact.

Explanation

Heine–Borel theorem.

Submit
Please wait...
About This Quiz
Advanced Compactness and Open Cover Applications - Quiz

Beyond the basics! This quiz brings complex problems and abstract reasoning about open covers. Try this quiz to challenge your grasp of compactness.

2)

What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) Which is compact?

Explanation

Closed and bounded.

Submit

3) Which open cover demonstrates compactness of [0,1]?

Explanation

Finite subcover exists.

Submit

4) A bounded open interval is compact.

Explanation

Open intervals are not compact.

Submit

5) Which is not compact in ℝ?

Explanation

Open intervals not compact.

Submit

6) Which property ensures compactness in ℝ^n?

Explanation

Closed and bounded sets are compact.

Submit

7) Which is an open cover of {0}?

Explanation

Any open set containing 0 works.

Submit

8) Compactness guarantees:

Explanation

Definition of compactness.

Submit

9) ℕ is compact in ℝ.

Explanation

ℕ is not bounded.

Submit

10) Which is compact in ℝ?

Explanation

Finite union of compact sets is compact.

Submit

11) Which is a counterexample to compactness?

Explanation

No finite subcover.

Submit

12) The union of two compact sets is compact.

Explanation

Union of compact sets is compact.

Submit

13) Which open cover proves {1/n : n∈ℕ} not compact without 0?

Explanation

Needs infinitely many sets.

Submit

14) Which statement is true?

Explanation

Heine–Borel theorem says compactness in ℝ means closed and bounded.

Submit
×
Saved
Thank you for your feedback!
View My Results
Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
Cancel
  • All
    All (14)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Every closed interval in ℝ is compact.
Which is compact?
Which open cover demonstrates compactness of [0,1]?
A bounded open interval is compact.
Which is not compact in ℝ?
Which property ensures compactness in ℝ^n?
Which is an open cover of {0}?
Compactness guarantees:
ℕ is compact in ℝ.
Which is compact in ℝ?
Which is a counterexample to compactness?
The union of two compact sets is compact.
Which open cover proves {1/n : n∈ℕ} not compact without 0?
Which statement is true?
play-Mute sad happy unanswered_answer up-hover down-hover success oval cancel Check box square blue
Alert!