Topological Continuity Quiz

1) In topology, a function is continuous if the preimage of every open set is open.

True

False

This is the standard topological definition of continuity.

Explanation

This is the standard topological definition of continuity.

2) In topology, continuity can be checked using open sets instead of limits.

True

False

Topological continuity does not depend on limits or metrics—only open sets.

Explanation

Topological continuity does not depend on limits or metrics—only open sets.

3) Every continuous function between topological spaces must also be continuous in the ε–δ sense.

True

False

ε–δ continuity only applies to metric spaces. Topological spaces need not be metric.

Explanation

ε–δ continuity only applies to metric spaces. Topological spaces need not be metric.

4) The preimage of a closed set under a continuous function is always closed.

True

False

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

Explanation

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

5) The preimage of an open set under any function is always open.

True

False

Only continuous functions guarantee this. Arbitrary functions do not.

Explanation

Only continuous functions guarantee this. Arbitrary functions do not.

6) A function between topological spaces is continuous at a point if every open neighborhood of the image contains the image of some open neighborhood of the point.

True

False

This is the pointwise topological definition of continuity.

Explanation

This is the pointwise topological definition of continuity.

7) A topological space must be a metric space for continuity to be defined.

True

False

Continuity is defined using open sets; no metric is required.

Explanation

Continuity is defined using open sets; no metric is required.

8) Which of the following is not guaranteed by topological continuity?

Images of compact sets are compact

Images of open sets are open

Preimages of open sets are open

Preimages of closed sets are closed

Continuous functions do not necessarily map open sets to open sets.

Explanation

Continuous functions do not necessarily map open sets to open sets.

9) If f : X → Y is continuous and X is connected, then:

F(X) must be dense

F(X) must be closed

F(X) must be connected

None of these

Continuous images of connected sets are always connected.

Explanation

Continuous images of connected sets are always connected.

10) If f : X → Y is not continuous, then:

There exists an open set in Y whose preimage is not open

F is not well-defined

F(X) is not compact

F(X) is not connected

Non-continuity means at least one open set fails the preimage-open condition.

Explanation

Non-continuity means at least one open set fails the preimage-open condition.

11) A function f : X → Y is continuous iff:

The image of every open set is open

The preimage of every closed set is closed

The image of every closed set is closed

The preimage of compact sets is compact

Continuity is equivalent to closed-set preimage condition.

Explanation

Continuity is equivalent to closed-set preimage condition.

12) Which is true about the composition of continuous functions?

Always continuous

Continuous only if injective

Continuous only if surjective

Never continuous

The composition of continuous functions is always continuous.

Explanation

The composition of continuous functions is always continuous.

13) Which statement is TRUE about continuity in topological spaces?

It can only be defined using sequences

It does not require distances

It requires metric structure to make sense

It requires a norm

Continuity uses only open sets, not distances or metrics.

Explanation

Continuity uses only open sets, not distances or metrics.

14) If f : X → Y is continuous and B ⊆ Y is closed, which must be true?

F⁻¹(B) is open

F⁻¹(B) is compact

F⁻¹(B) is connected

F⁻¹(B) is closed

Continuous functions pull back closed sets to closed sets.

Explanation

Continuous functions pull back closed sets to closed sets.

15) The condition “images of connected sets are connected” holds for:

All functions

Continuous functions only

Functions between compact spaces only

Functions between Hausdorff spaces only

Only continuous functions preserve connectedness.

Explanation

Only continuous functions preserve connectedness.

Topological Continuity Quiz

1) In topology, a function is continuous if the preimage of every open set is open.

2)

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

2) In topology, continuity can be checked using open sets instead of limits.

3) Every continuous function between topological spaces must also be continuous in the ε–δ sense.

4) The preimage of a closed set under a continuous function is always closed.

5) The preimage of an open set under any function is always open.

6) A function between topological spaces is continuous at a point if every open neighborhood of the image contains the image of some open neighborhood of the point.

7) A topological space must be a metric space for continuity to be defined.

8) Which of the following is not guaranteed by topological continuity?

9) If f : X → Y is continuous and X is connected, then:

10) If f : X → Y is not continuous, then:

11) A function f : X → Y is continuous iff:

12) Which is true about the composition of continuous functions?

13) Which statement is TRUE about continuity in topological spaces?

14) If f : X → Y is continuous and B ⊆ Y is closed, which must be true?

15) The condition “images of connected sets are connected” holds for: