Topological Continuity Quiz

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| Questions: 15 | Updated: Nov 24, 2025
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1) In topology, a function is continuous if the preimage of every open set is open.

Explanation

This is the standard topological definition of continuity.

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About This Quiz
Topological Continuity Quiz - Quiz

Ready to understand continuity beyond ε–δ definitions? This quiz introduces the topological definition of continuity, where open sets and their preimages determine whether a function is continuous. You’ll explore how continuous functions behave with open and closed sets, how connectedness is preserved, and when continuity fails. By working through examples... see moreinvolving topological spaces, compactness, and function composition, you’ll develop a deeper understanding of how continuity is defined in general topology. By the end, you’ll confidently evaluate continuous functions using open-set methods! see less

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

Explanation

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

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

Explanation

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

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

Explanation

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

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

Explanation

Only continuous functions guarantee this. Arbitrary functions do not.

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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.

Explanation

This is the pointwise topological definition of continuity.

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

Explanation

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

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

Explanation

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

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

Explanation

Continuous images of connected sets are always connected.

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

Explanation

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

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

Explanation

Continuity is equivalent to closed-set preimage condition.

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

Explanation

The composition of continuous functions is always continuous.

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

Explanation

Continuity uses only open sets, not distances or metrics.

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

Explanation

Continuous functions pull back closed sets to closed sets.

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

Explanation

Only continuous functions preserve connectedness.

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In topology, a function is continuous if the preimage of every open...
In topology, continuity can be checked using open sets instead of...
Every continuous function between topological spaces must also be...
The preimage of a closed set under a continuous function is always...
The preimage of an open set under any function is always open.
A function between topological spaces is continuous at a point if...
A topological space must be a metric space for continuity to be...
Which of the following is not guaranteed by topological continuity?
If f : X → Y is continuous and X is connected, then:
If f : X → Y is not continuous, then:
A function f : X → Y is continuous iff:
Which is true about the composition of continuous functions?
Which statement is TRUE about continuity in topological spaces?
If f : X → Y is continuous and B ⊆ Y is closed, which must be...
The condition “images of connected sets are connected” holds for:
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