Understanding Open Covers

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7060 | Total Attempts: 9,520,935
| Questions: 15 | Updated: Oct 13, 2025
Please wait...
Question 1 / 15
0 %
0/100
Score 0/100
1) An open cover of a set is:

Explanation

An open cover is a family of open sets whose union contains the set.

Submit
Please wait...
About This Quiz
Understanding Open Covers - Quiz

Covers and subsets can feel abstract. In this quiz, you’ll get comfortable identifying open covers and their role in compactness. Take this quiz to build foundational intuition.

2)
We’ll put your name on your report, certificate, and leaderboard.
2) Which is an open cover of [0,1]?

Explanation

These two open intervals together cover [0,1].

Submit
3) Every finite set in ℝ is compact.

Explanation

Finite sets are always compact because any cover has a finite subcover.

Submit
4) Which of these is an open set in ℝ?

Explanation

Open intervals exclude endpoints.

Submit
5) Which is not an open cover of (0,1)?

Explanation

(−1,2) contains closed endpoints, not an open cover.

Submit
6) A finite subcover is:

Explanation

Compactness means every open cover has a finite subcover.

Submit
7) True/False: An open cover may consist of infinitely many sets.

Explanation

Covers can be infinite collections.

Submit
8) The union of sets in an open cover is:

Explanation

The union must contain the set fully.

Submit
9) Which of the following is an open cover of (0,1)?

Explanation

Together these intervals cover (0,1).

Submit
10) Open covers can overlap.

Explanation

Covers often overlap; disjointness is not required.

Submit
11) Compactness in ℝⁿ relates to:

Explanation

By Heine–Borel, compact = closed + bounded.

Submit
12) Which is compact in ℝ?

Explanation

[0,1] is both closed and bounded.

Submit
13) (0,1) is compact.

Explanation

Open intervals are not compact.

Submit
14) Which of these is not compact?

Explanation

(0,1) is open, hence not compact.

Submit
15) Compactness ensures:

Explanation

Definition of compactness.

Submit
View My Results
Cancel
  • All
    All (15)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
An open cover of a set is:
Which is an open cover of [0,1]?
Every finite set in ℝ is compact.
Which of these is an open set in ℝ?
Which is not an open cover of (0,1)?
A finite subcover is:
True/False: An open cover may consist of infinitely many sets.
The union of sets in an open cover is:
Which of the following is an open cover of (0,1)?
Open covers can overlap.
Compactness in ℝⁿ relates to:
Which is compact in ℝ?
(0,1) is compact.
Which of these is not compact?
Compactness ensures:
Alert!

Back to Top Back to top
Advertisement