Preimage Of Open Sets Quiz

1) If f:X→Y is continuous, then for every open set U⊆Y, the preimage f^{-1}(U) is open in X.

A. True

B. False

This is the definition of continuity in topological terms.

Explanation

This is the definition of continuity in topological terms.

2) For any function f:X→Y, f^{-1}(∅)=∅.

A. True

B. False

Preimage of empty set is empty.

Explanation

Preimage of empty set is empty.

3) For any function f:X→Y, f^{-1}(Y)=X.

A. True

B. False

Everything maps into Y.

Explanation

Everything maps into Y.

4) For sets A,B⊆Y, f^{-1}(A∩B)=f^{-1}(A)∩f^{-1}(B).

A. True

B. False

Preimages preserve intersections.

Explanation

Preimages preserve intersections.

5) For sets A,B⊆Y, f^{-1}(A∪B)=f^{-1}(A)∪f^{-^{-1}(B)}.

A. True

B. False

Preimages preserve unions.

Explanation

Preimages preserve unions.

6) If U⊆Y is open and f is any function, then f^{-1}(U) must be open in X.

A. True

B. False

Only continuous functions preserve openness.

Explanation

Only continuous functions preserve openness.

7) If f is continuous, the preimage of every closed set in Y is closed in X.

A. True

B. False

Equivalent definition of continuity.

Explanation

Equivalent definition of continuity.

8) If the preimage of every open set in Y is open in X, then f must be continuous.

A. True

B. False

This is the definition.

Explanation

This is the definition.

9) Which can be used to test continuity of f?

A. Preimage of open sets open

B. Preimage of closed sets closed

C. Preimage of basis elements open

D. Images of open sets open

Images of open sets need not be open.

Explanation

Images of open sets need not be open.

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10) If f is continuous and C⊆Y is closed, then:

A. f^{-1}(C) closed

B. f^{-1}(C^c) open

C. f^{-1}(C) compact

D. f^{-1}(C^c)=X\f^{-1}(C)

Preimages preserve closedness and complements.

Explanation

Preimages preserve closedness and complements.

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11) Preimages preserve which set operations?

A. Union

B. Intersection

C. Complement

D. Set difference

Preimages preserve all basic operations.

Explanation

Preimages preserve all basic operations.

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12) If U⊆Y is open and f is continuous, then f^{-1}(U) is:

A. Always open

B. Always closed

C. Sometimes compact

D. Always empty

By continuity.

Explanation

By continuity.

13) Let f:X→Y be continuous. Which sets have closed preimages?

A. Closed sets

B. Finite unions of closed sets

C. Interiors of closed sets

D. Arbitrary intersections of closed sets

Closed sets, finite unions, intersections remain closed.

Explanation

Closed sets, finite unions, intersections remain closed.

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14) If A⊆Y and f^{-1}(A)=∅, which must be true?

A. A=∅

B. A⊆f(X)

C. A∩f(X)=∅

D. f is constant

A has no point in f(X).

Explanation

A has no point in f(X).

15) If the preimage of every basis element of Y is open, then:

A. f is open

B. f is continuous

C. f is closed

D. f injective

Preimages of basis elements determine continuity.

Explanation

Preimages of basis elements determine continuity.

Preimage of Open Sets Quiz

1) If f:X→Y is continuous, then for every open set U⊆Y, the preimage f^{-1}(U) is open in X.

2)

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

2) For any function f:X→Y, f^{-1}(∅)=∅.

3) For any function f:X→Y, f^{-1}(Y)=X.

4) For sets A,B⊆Y, f^{-1}(A∩B)=f^{-1}(A)∩f^{-1}(B).

5) For sets A,B⊆Y, f^{-1}(A∪B)=f^{-1}(A)∪f^{-^{-1}(B)}.

6) If U⊆Y is open and f is any function, then f^{-1}(U) must be open in X.

7) If f is continuous, the preimage of every closed set in Y is closed in X.

8) If the preimage of every open set in Y is open in X, then f must be continuous.

9) Which can be used to test continuity of f?

10) If f is continuous and C⊆Y is closed, then:

11) Preimages preserve which set operations?

12) If U⊆Y is open and f is continuous, then f^{-1}(U) is:

13) Let f:X→Y be continuous. Which sets have closed preimages?

14) If A⊆Y and f^{-1}(A)=∅, which must be true?

15) If the preimage of every basis element of Y is open, then: