Applying Topological Continuity Quiz

1) A function f : X → Y is continuous if:

The preimage of every open set in Y is open in X.

The preimage of every closed set in Y is closed in X.

For each x ∈ X, the function preserves neighborhood structure.

F is injective.

All three are equivalent definitions of continuity in topology.

Explanation

All three are equivalent definitions of continuity in topology.

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

F⁻¹(U) is open

F⁻¹(U) is closed

F⁻¹(U) may be empty

F⁻¹(U) = X

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

Explanation

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

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3) Which imply continuity at a point x₀?

Every neighborhood of f(x₀) has a preimage neighborhood of x₀

Preimages of open sets containing f(x₀) are open

Sequential convergence is preserved in metric spaces

Function is constant

All three describe continuity; constancy is not required.

Explanation

All three describe continuity; constancy is not required.

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4) F : X → Y is continuous if:

It preserves limits of sequences (metric spaces)

Preimages of basis elements of Y are open

It maps connected sets to connected sets

It maps compact sets to open sets

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

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:

Preimage of any open set is itself

It preserves neighborhood structure

It is bijective

It preserves connectedness

Identity map preserves open sets and neighborhoods.

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:

All subsets of finite sets are cofinite

All subsets of X are open

Preimages of open sets in Y are open in X

X is discrete

Finite cofinite topology becomes discrete.

Explanation

Finite cofinite topology becomes discrete.

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

Every function f : X → Y is continuous

Only constant functions are continuous

Only injective functions are continuous

Preimages of open sets are automatically open

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

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:

F is continuous

F is open

Preimages of arbitrary open sets are open

F is injective

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

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:

F(K) is compact

K must be closed

Images of compact sets preserve compactness

F(K) is open

Continuous images of compact sets are compact.

Explanation

Continuous images of compact sets are compact.

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

Preimages of neighborhoods

Limits of nets

Preimages of basis neighborhoods

Compactness of image

All characterize continuity; compactness is unrelated.

Explanation

All characterize continuity; compactness is unrelated.

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

F can be any function

F is automatically continuous

Preimages of open sets in Y are open

Only constant functions are continuous

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

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:

ε–δ definition

Sequential convergence

Open-set preimages

Closed-set preimages

All four definitions are equivalent.

Explanation

All four definitions are equivalent.

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

Its image is a single point

The codomain is discrete

The image is connected but finite

The function is continuous

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

Explanation

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

14) Which implies discontinuity?

Preimage of an open set is not open

There exists one open set whose preimage fails to be open

Images of compact sets are not compact

Images of connected sets are not connected

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

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?

F(x) = constant

F(x) = identity (if X = Y)

Any function

Only surjective functions

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

Explanation

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

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Applying Topological Continuity Quiz

1) A function f : X → Y is continuous if:

2)

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

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

3) Which imply continuity at a point x₀?

4) F : X → Y is continuous if:

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

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

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

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

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

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

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

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

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

14) Which implies discontinuity?

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