Preimage Of Open Sets AppliCATion Quiz

1) If f is continuous, the preimage of a finite union of closed sets is closed.

True

False

Preimages preserve unions, and continuous functions preserve closedness → finite unions of closed sets pull back to closed sets.

Explanation

Preimages preserve unions, and continuous functions preserve closedness → finite unions of closed sets pull back to closed sets.

2) The preimage of a singleton under a surjective function is always nonempty.

True

False

Surjective means every element of Y has at least one preimage in X.

Explanation

Surjective means every element of Y has at least one preimage in X.

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

True

False

For open sets U = int(U). Their preimages are identical.

Explanation

For open sets U = int(U). Their preimages are identical.

4) For any function f and A ⊆ Y, f(f^{-1}(A)) ⊆ A.

True

False

Applying f after taking the preimage cannot give elements outside A.

Explanation

Applying f after taking the preimage cannot give elements outside A.

5) If f is continuous and A ⊆ Y is closed, then f^{-1}(A) is closed in X.

True

False

Closed-set preimages remain closed—equivalent to open-set definition.

Explanation

Closed-set preimages remain closed—equivalent to open-set definition.

6) If f is continuous, then for every x∈X there exists an open U⊆Y such that x∈f^{-1}(U).

True

False

Take U = Y (open). Then f^{-1}(Y)=X, which contains all x.

Explanation

Take U = Y (open). Then f^{-1}(Y)=X, which contains all x.

7) If f^{-1}(U) is open for every basis element U of Y, then f is continuous.

True

False

Checking preimages on a basis is enough to guarantee continuity.

Explanation

Checking preimages on a basis is enough to guarantee continuity.

8) If f^{-1}(A) is open and A ⊆ Y, then A must be open in Y.

True

False

A non-open set can pull back to an open one.

Explanation

A non-open set can pull back to an open one.

9) The preimage operation is monotone: A⊆B ⇒ f^{-1}(A)⊆f^{-1}(B).

True

False

If A is smaller, its preimage is also smaller.

Explanation

If A is smaller, its preimage is also smaller.

10) If f is continuous, x_n→x in X and y_n=f(x_n), then every limit point of (y_n) lies in f(x).

True

False

Continuity implies f(x_n)→f(x); all subsequential limits equal f(x).

Explanation

Continuity implies f(x_n)→f(x); all subsequential limits equal f(x).

11) The preimage of an open set under a continuous function is always:

Closed

Open

Compact

Connected

This is the definition of continuity.

Explanation

This is the definition of continuity.

12) Which statements about preimages are always true for any function f : X→Y?

F^{-1}(A∩B) = f^{-1}(A) ∩ f^{-1}(B)

F^{-1}(∅) = ∅

F^{-1}(Y) = X

F^{-1}(A) must be open only if A is open

Preimages always preserve intersections, the empty set, and whole spaces.

Explanation

Preimages always preserve intersections, the empty set, and whole spaces.

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13) If f is not continuous, which of the following can still be true?

All preimages of open sets are open

Some preimages of open sets are open

No preimage of any open set is open

Preimages of open sets are always closed

Non-continuous functions may sometimes have open preimages, or none.

Explanation

Non-continuous functions may sometimes have open preimages, or none.

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14) Which conditions characterize continuity of f : X→Y?

Preimage of every open set is open

Preimage of every closed set is closed

Image of every open set is open

Preimage of every basis element is open

These are equivalent definitions of continuity.

Explanation

These are equivalent definitions of continuity.

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15) Let U⊆Y be open. f^{-1}(U) is:

The set of all y∈Y that map into U

The set of all x∈X whose image lies in U

The image of U under f

The complement of U

Preimage = all x whose f(x) lies in U.

Explanation

Preimage = all x whose f(x) lies in U.

Preimage of Open Sets Application Quiz

1) If f is continuous, the preimage of a finite union of closed sets is closed.

2)

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

2) The preimage of a singleton under a surjective function is always nonempty.

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

4) For any function f and A ⊆ Y, f(f^{-1}(A)) ⊆ A.

5) If f is continuous and A ⊆ Y is closed, then f^{-1}(A) is closed in X.

6) If f is continuous, then for every x∈X there exists an open U⊆Y such that x∈f^{-1}(U).

7) If f^{-1}(U) is open for every basis element U of Y, then f is continuous.

8) If f^{-1}(A) is open and A ⊆ Y, then A must be open in Y.

9) The preimage operation is monotone: A⊆B ⇒ f^{-1}(A)⊆f^{-1}(B).

10) If f is continuous, x_n→x in X and y_n=f(x_n), then every limit point of (y_n) lies in f(x).

11) The preimage of an open set under a continuous function is always:

12) Which statements about preimages are always true for any function f : X→Y?

13) If f is not continuous, which of the following can still be true?

14) Which conditions characterize continuity of f : X→Y?

15) Let U⊆Y be open. f^{-1}(U) is: