Applying Topological Continuity Quiz

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| Questions: 15 | Updated: Nov 24, 2025
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1) A function f : X → Y is continuous if:

Explanation

All three are equivalent definitions of continuity in topology.

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About This Quiz
Applying Topological Continuity Quiz - Quiz

How well can you apply the topological definition of continuity? This quiz helps you connect theory to practice by examining preimages of open sets, behavior on basis elements, continuity in discrete and trivial topologies, compactness preservation, and identity maps. You’ll analyze how continuous functions behave in different spaces and learn... see morewhen discontinuity occurs through preimage failures. By the end, you’ll confidently apply the open-set definition of continuity across a wide range of topological situations! see less

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2) If f : X → Y is continuous and U ⊆ Y is open, then:

Explanation

Preimages of open sets are open; they may also be empty.

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3) Which imply continuity at a point x₀?

Explanation

All three describe continuity; constancy is not required.

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4) F : X → Y is continuous if:

Explanation

Sequence-limit preservation works in metric spaces; basis preimages work in topological spaces.

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5) The identity map id : X → X is continuous because:

Explanation

Identity map preserves open sets and neighborhoods.

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6) If X is finite and has the cofinite topology, every function from X to Y is continuous because:

Explanation

Finite cofinite topology becomes discrete.

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7) If the codomain Y has the trivial topology, then:

Explanation

Only ∅ and Y are open; their preimages are always open.

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8) Suppose f : X → Y and every preimage of a basis element of Y is open. Then:

Explanation

Preimages of basis elements open ⇒ preimages of arbitrary open sets (unions) are open.

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9) If f is continuous and K ⊆ X is compact, then:

Explanation

Continuous images of compact sets are compact.

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10) To check continuity at a point, we may verify:

Explanation

All characterize continuity; compactness is unrelated.

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11) If f : X → Y is continuous and X is discrete:

Explanation

All subsets of a discrete space are open, so any function is continuous.

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12) In a metric space, continuity can be checked using:

Explanation

All four definitions are equivalent.

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13) A function from a connected domain is constant if:

Explanation

A function is constant iff its image is a single point.

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14) Which implies discontinuity?

Explanation

Failure of the open-set preimage condition implies non-continuity.

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15) Let X have the trivial topology and Y be arbitrary. Which are continuous?

Explanation

All preimages of ∅ and X are open, so all functions are continuous.

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A function f : X → Y is continuous if:
If f : X → Y is continuous and U ⊆ Y is open, then:
Which imply continuity at a point x₀?
F : X → Y is continuous if:
The identity map id : X → X is continuous because:
If X is finite and has the cofinite topology, every function from X to...
If the codomain Y has the trivial topology, then:
Suppose f : X → Y and every preimage of a basis element of Y is...
If f is continuous and K ⊆ X is compact, then:
To check continuity at a point, we may verify:
If f : X → Y is continuous and X is discrete:
In a metric space, continuity can be checked using:
A function from a connected domain is constant if:
Which implies discontinuity?
Let X have the trivial topology and Y be arbitrary. Which are...
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