Preimage Behavior In Continuous Functions Quiz

1) If f is continuous and U ⊆ Y is path-connected, then f^{-1}(U) must be path-connected.

True

False

Continuity does not preserve path-connectedness in preimages; only images of connected sets remain connected.

Explanation

Continuity does not preserve path-connectedness in preimages; only images of connected sets remain connected.

2) For a bijection f: X → Y, continuity of f is equivalent to openness of preimages of open sets.

True

False

Continuity means the preimage of every open set is open; bijectivity is irrelevant.

Explanation

Continuity means the preimage of every open set is open; bijectivity is irrelevant.

3) If f: X → Y and g: Y → Z are functions, (g∘f)^{-1}(W) = f^{-1}(g^{-1}(W)).

True

False

Preimages respect function composition.

Explanation

Preimages respect function composition.

4) If A ⊆ Y is dense, then f^{-1}(A) is dense in X for every continuous f.

True

False

Density does not necessarily pull back under continuous maps.

Explanation

Density does not necessarily pull back under continuous maps.

5) If f is continuous and U ⊆ Y is open, then f(f^{-1}(U)) = U.

True

False

Only f(f^{-1}(U)) ⊆ U is guaranteed; equality need not hold.

Explanation

Only f(f^{-1}(U)) ⊆ U is guaranteed; equality need not hold.

6) The preimage of a compact set under a continuous function is:

Always compact

Sometimes compact, sometimes not

Always open

Always closed

Compactness is preserved in images, not preimages.

Explanation

Compactness is preserved in images, not preimages.

7) Which functions are guaranteed to have open preimages of open sets?

Identity functions

Continuous functions

Constant functions

Arbitrary functions

This is exactly the definition of continuity.

Explanation

This is exactly the definition of continuity.

8) For any function f: X → Y, the preimage of a singleton {y}:

Must be open

Must be closed

May be any subset of X

Must contain exactly one point

A preimage of a point can be any subset of X.

Explanation

A preimage of a point can be any subset of X.

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

Check preimages of open sets are open

Check preimages of closed sets are closed

Check preimages of basis elements are open

Check images of open sets are open

All three are valid equivalent continuity conditions.

Explanation

All three are valid equivalent continuity conditions.

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

F^{-1}(C) is closed

F^{-1}(C^c) is open

F^{-1}(C) must be compact

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

Closed-set preimages remain closed; complements pull back correctly.

Explanation

Closed-set preimages remain closed; complements pull back correctly.

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11) Preimages preserve which set operations?

Union

Intersection

Complement

Set difference

Preimages preserve all basic set operations.

Explanation

Preimages preserve all basic set operations.

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

Always open

Always closed

Sometimes compact

Always empty

This is the definition of continuity.

Explanation

This is the definition of continuity.

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

Closed sets in Y

Finite unions of closed sets in Y

Interiors of closed sets in Y

Arbitrary intersections of closed sets in Y

Preimages respect intersections and unions of closed sets.

Explanation

Preimages respect intersections and unions of closed sets.

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

A = ∅

A ∩ f(X) = ∅

A ⊆ f(X)

F is constant

If no x maps into A, then A contains no point of the image of f.

Explanation

If no x maps into A, then A contains no point of the image of f.

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

F is an open map

F is continuous

F is closed

F must be injective

Open preimages of basis elements are enough to guarantee continuity.

Explanation

Open preimages of basis elements are enough to guarantee continuity.

Preimage Behavior in Continuous Functions Quiz

1) If f is continuous and U ⊆ Y is path-connected, then f^{-1}(U) must be path-connected.

2)

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

2) For a bijection f: X → Y, continuity of f is equivalent to openness of preimages of open sets.

3) If f: X → Y and g: Y → Z are functions, (g∘f)^{-1}(W) = f^{-1}(g^{-1}(W)).

4) If A ⊆ Y is dense, then f^{-1}(A) is dense in X for every continuous f.

5) If f is continuous and U ⊆ Y is open, then f(f^{-1}(U)) = U.

6) The preimage of a compact set under a continuous function is:

7) Which functions are guaranteed to have open preimages of open sets?

8) For any function f: X → Y, the preimage of a singleton {y}:

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 must have closed preimages?

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

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