Topological Continuity Concepts Quiz

1) Which of the following is the correct topological definition of continuity of f : X → Y?

Images of open sets are open

Preimages of open sets are open

Images of closed sets are closed

Preimages of compact sets are compact

In topology, continuity is defined by open-set preimages being open.

Explanation

In topology, continuity is defined by open-set preimages being open.

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

Always open

Always closed

Either open or closed

Not defined

This is the core property of continuous functions in topology.

Explanation

This is the core property of continuous functions in topology.

3) A function f : X → Y is continuous at a point x ∈ X if:

Every image neighborhood is open

Every preimage of a neighborhood of f(x) contains a neighborhood of x

Both A and B

None of the above

Continuity at a point means neighborhoods of f(x) pull back to neighborhoods of x.

Explanation

Continuity at a point means neighborhoods of f(x) pull back to neighborhoods of x.

4) A function from a discrete space into any topological space is:

Always discontinuous

Always continuous

Continuous only if injective

Continuous only if surjective

In a discrete domain, all subsets are open → all preimages are open → continuity is automatic.

Explanation

In a discrete domain, all subsets are open → all preimages are open → continuity is automatic.

5) The identity map id : X → X is:

Continuous for some topologies

Continuous only for metric spaces

Always continuous

Never continuous

The identity map preserves open sets exactly → always continuous.

Explanation

The identity map preserves open sets exactly → always continuous.

6) If f : X → Y is continuous, then:

Preimages of closed sets are closed

Images of open sets are always open

Images of closed sets are always closed

None of the above

Closed-set preimage continuity is equivalent to open-set continuity.

Explanation

Closed-set preimage continuity is equivalent to open-set continuity.

7) Which of the following guarantees a function is continuous?

The codomain is discrete

The domain is discrete

The codomain is compact

Both A and B

If the domain is discrete, all preimages are open; if the codomain is discrete, every set is open, so preimages are open.

Explanation

If the domain is discrete, all preimages are open; if the codomain is discrete, every set is open, so preimages are open.

8) A constant function is always continuous in any topological spaces.

True

False

The preimage of any open set is either ∅ or the whole domain—both open.

Explanation

The preimage of any open set is either ∅ or the whole domain—both open.

9) If a function between topological spaces is not continuous at a point, then there exists an open set in the codomain whose preimage is not open.

True

False

Non-continuity means at least one open-set preimage fails to be open.

Explanation

Non-continuity means at least one open-set preimage fails to be open.

10) If f : X → Y is continuous and B ⊆ Y is open, then f(B) must be open in Y.

True

False

Continuity says nothing about images of open sets—only preimages.

Explanation

Continuity says nothing about images of open sets—only preimages.

11) A function is continuous if and only if the preimage of every closed set is closed.

True

False

Closed-set and open-set definitions of continuity are equivalent.

Explanation

Closed-set and open-set definitions of continuity are equivalent.

12) If the domain has the trivial (indiscrete) topology, every function from it is continuous.

True

False

The only open sets are ∅ and X. Their preimages are always open → all functions are continuous.

Explanation

The only open sets are ∅ and X. Their preimages are always open → all functions are continuous.

13) If f : X → Y is continuous and A ⊆ X is open, then f|ₐ (the restriction) is always continuous.

True

False

Restricting a continuous function preserves the open-set preimage property.

Explanation

Restricting a continuous function preserves the open-set preimage property.

14) If f : X → Y is continuous, then f⁻¹(∅) = ∅ is open.

True

False

The preimage of the empty set is empty—and ∅ is always open.

Explanation

The preimage of the empty set is empty—and ∅ is always open.

15) A function can be continuous at some points and discontinuous at others in topological spaces.

True

False

Pointwise continuity allows a function to behave differently at different points.

Explanation

Pointwise continuity allows a function to behave differently at different points.