Finite Subcovers Quiz

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Quizzes Created: 7387 | Total Attempts: 9,536,020
| Questions: 15 | Updated: Nov 24, 2025
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1) An open cover of a set X is a collection of open sets whose union contains X.

Explanation

This is the definition of an open cover: many open sets whose union includes the whole set.

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About This Quiz
Finite Subcovers Quiz - Quiz

Ready to understand one of the most important ideas in topology? This quiz helps you explore compactness through the lens of open covers and finite subcovers. You’ll work with classic examples like closed intervals, bounded sets, and subsets of ℝⁿ to see why some sets are compact and others fail... see moreto be. By testing properties such as closed-subset compactness, continuous images of compact sets, and Hausdorff behavior, you’ll build a clear understanding of how compactness shapes the structure of topological spaces. By the end, you’ll be confident identifying compact sets and explaining why finite subcovers matter! see less

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2) A finite subcover is any finite subcollection of a cover.

Explanation

A finite subcover must still cover X. A random finite subcollection might not.

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3) A topological space is compact if and only if every open cover has a finite subcover.

Explanation

This is the formal definition of compactness.

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4) If there exists at least one open cover of a space that has a finite subcover, then the space is compact.

Explanation

Compactness requires every open cover to have a finite subcover, not just one.

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5) The closed interval [0,1] ⊆ ℝ is compact because every open cover of it has a finite subcover.

Explanation

By the Heine–Borel theorem, closed and bounded intervals in ℝ are compact.

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6) The open interval (0,1) ⊆ ℝ is not compact because there exists an open cover with no finite subcover.

Explanation

Example cover: {(1/n,1) : n∈N} has no finite subcover.

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7) Every closed subset of a compact space is compact.

Explanation

Closed subsets inherit compactness.

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8) Every open subset of a compact space is compact.

Explanation

Open subsets may fail to be compact.

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9) The continuous image of a compact space is compact.

Explanation

Continuity preserves compactness under images.

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10) In any Hausdorff space, compact subsets are closed.

Explanation

Compact sets in Hausdorff spaces are closed.

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11) Which best defines an open cover of a set X in a topological space?

Explanation

Union must contain X; not necessarily equal or finite.

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12) What is a finite subcover?

Explanation

A subcover must continue to cover the whole set.

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13) A topological space is compact if and only if:

Explanation

Compactness is defined using open covers.

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14) In ℝⁿ with the usual topology, which condition characterizes compact sets?

Explanation

Heine–Borel theorem.

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15) Which of the following subsets of ℝ is compact?

Explanation

[0,1] is closed and bounded.

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An open cover of a set X is a collection of open sets whose union...
A finite subcover is any finite subcollection of a cover.
A topological space is compact if and only if every open cover has a...
If there exists at least one open cover of a space that has a finite...
The closed interval [0,1] ⊆ ℝ is compact because every open cover...
The open interval (0,1) ⊆ ℝ is not compact because there exists an...
Every closed subset of a compact space is compact.
Every open subset of a compact space is compact.
The continuous image of a compact space is compact.
In any Hausdorff space, compact subsets are closed.
Which best defines an open cover of a set X in a topological space?
What is a finite subcover?
A topological space is compact if and only if:
In ℝⁿ with the usual topology, which condition characterizes...
Which of the following subsets of ℝ is compact?
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