Preimage of Open Sets Quiz

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| Questions: 15 | Updated: Dec 12, 2025
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1) If f:X→Y is continuous, then for every open set U⊆Y, the preimage f^{-1}(U) is open in X.

Explanation

This is the definition of continuity in topological terms.

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About This Quiz
Preimage Of Open Sets Quiz - Quiz

Ready to deepen your understanding of continuity in topology? This quiz focuses on the foundational principle that continuous functions pull back open sets to open sets. You’ll analyze how preimages behave under unions, intersections, complements, and singletons, and explore how continuity connects to compactness and connectedness through preimages. By working... see morethrough these problems, you'll develop a strong intuition for how functions interact with open sets, and why preimage behavior forms the backbone of topological continuity. By the end, you'll be ready to evaluate continuity using preimages with confidence!
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2) For any function f:X→Y, f^{-1}(∅)=∅.

Explanation

Preimage of empty set is empty.

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

Explanation

Everything maps into Y.

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

Explanation

Preimages preserve intersections.

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

Explanation

Preimages preserve unions.

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

Explanation

Only continuous functions preserve openness.

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

Explanation

Equivalent definition of continuity.

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

Explanation

This is the definition.

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

Explanation

Images of open sets need not be open.

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

Explanation

Preimages preserve closedness and complements.

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11) Preimages preserve which set 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:

Explanation

By continuity.

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

Explanation

Closed sets, finite unions, intersections remain closed.

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

Explanation

A has no point in f(X).

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

Explanation

Preimages of basis elements determine continuity.

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If f:X→Y is continuous, then for every open set U⊆Y, the preimage...
For any function f:X→Y, f^{-1}(∅)=∅.
For any function f:X→Y, f^{-1}(Y)=X.
For sets A,B⊆Y, f^{-1}(A∩B)=f^{-1}(A)∩f^{-1}(B).
For sets A,B⊆Y, f^{-1}(A∪B)=f^{-1}(A)∪f^{-^{-1}(B)}.
If U⊆Y is open and f is any function, then f^{-1}(U) must be open in...
If f is continuous, the preimage of every closed set in Y is closed in...
If the preimage of every open set in Y is open in X, then f must be...
Which can be used to test continuity of f?
If f is continuous and C⊆Y is closed, then:
Preimages preserve which set operations?
If U⊆Y is open and f is continuous, then f^{-1}(U) is:
Let f:X→Y be continuous. Which sets have closed preimages?
If A⊆Y and f^{-1}(A)=∅, which must be true?
If the preimage of every basis element of Y is open, then:
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