Sequential Compactness Quiz

1) A space X is sequentially compact if every sequence in X has a convergent subsequence whose limit is in X.

True

False

Definition of sequential compactness.

Explanation

Definition of sequential compactness.

2) If a sequence in a sequentially compact space has no convergent subsequence, then the space is not sequentially compact.

True

False

Violates definition.

Explanation

Violates definition.

3) In ℝ, every closed and bounded set is sequentially compact.

True

False

Heine-Borel + Bolzano-Weierstrass.

Explanation

Heine-Borel + Bolzano-Weierstrass.

4) Every sequentially compact set must be bounded in ℝ.

True

False

Otherwise sequences diverge.

Explanation

Otherwise sequences diverge.

5) Every compact set is sequentially compact in all topological spaces.

True

False

Not equivalent in general spaces.

Explanation

Not equivalent in general spaces.

6) Which defines sequential compactness?

Every open cover has finite subcover

Every sequence has convergent subsequence

Every subsequence converges

Set closed but not bounded

Sequential definition.

Explanation

Sequential definition.

7) In a metric space sequential compactness is equivalent to:

Total disconnectedness

Compactness

Openness

Countability

Metric equivalence.

Explanation

Metric equivalence.

8) Which is sequentially compact in ℝ?

(0,1)

[0,1]

[0,∞)

(−∞,0]

Closed and bounded.

Explanation

Closed and bounded.

9) Suppose X sequentially compact. Which must be true?

Every sequence converges

Every sequence has conv subseq

X finite

X open

Definition.

Explanation

Definition.

10) A sequence in sequential compact space:

Always converges

Must have bounded subseq

Must have convergent subseq

Cannot be constant

Definition.

Explanation

Definition.

11) Which true about sequential compactness in ℝ?

Implies bounded

Implies closed

Sequence must be monotone

Every seq has conv subseq

Sequential compact ⇔ compact.

Explanation

Sequential compact ⇔ compact.

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12) A set in ℝ fails sequential compact if:

Unbounded

Not closed

Has seq w/o conv subseq

Finite

All break closure/boundedness.

Explanation

All break closure/boundedness.

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13) Properties guaranteed in metric spaces:

Totally bounded

Closed

Countable

Every seq has conv subseq

Sequentially compact ⇔ compact.

Explanation

Sequentially compact ⇔ compact.

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14) Which sequences show failure in (0,1)?

1/n

1−1/n

N mod 1

N/(n+1)

Dense no conv subseq.

Explanation

Dense no conv subseq.

15) Equivalent in metric spaces:

Every seq has Cauchy subseq

Complete+totally bounded

Every open cover finite subcover

Every seq has conv subseq

All equivalent.

Explanation

All equivalent.

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Sequential Compactness Quiz

1) A space X is sequentially compact if every sequence in X has a convergent subsequence whose limit is in X.

2)

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

2) If a sequence in a sequentially compact space has no convergent subsequence, then the space is not sequentially compact.

3) In ℝ, every closed and bounded set is sequentially compact.

4) Every sequentially compact set must be bounded in ℝ.

5) Every compact set is sequentially compact in all topological spaces.

6) Which defines sequential compactness?

7) In a metric space sequential compactness is equivalent to:

8) Which is sequentially compact in ℝ?

9) Suppose X sequentially compact. Which must be true?

10) A sequence in sequential compact space:

11) Which true about sequential compactness in ℝ?

12) A set in ℝ fails sequential compact if:

13) Properties guaranteed in metric spaces:

14) Which sequences show failure in (0,1)?

15) Equivalent in metric spaces: