Interior Points Properties Quiz

1) A point x is an interior point of a set U if:

X is a limit of U

Every open ball around x intersects U

There exists an open ball around x contained in U

X is in the closure of U

Interior means there is a small open ball completely inside the set.

Explanation

Interior means there is a small open ball completely inside the set.

2) The interior of the interval (0,1) is:

(0,1)

[0,1]

{0,1}

∅

(0,1) is already open, so its interior is itself.

Explanation

(0,1) is already open, so its interior is itself.

3) The interior of the interval [0,1] is:

(0,1)

[0,1]

{0,1}

∅

The interior points are those with a small open interval inside [0,1], which is (0,1).

Explanation

The interior points are those with a small open interval inside [0,1], which is (0,1).

4) The interior of a set is always

Closed

Bounded

Finite

Open

The interior is defined as the largest open set inside a set.

Explanation

The interior is defined as the largest open set inside a set.

5) Which set has no interior in ℝ?

(0,1)

[0,1]

{0,1}

{1}

Finite sets have no intervals inside them, so they have no interior points.

Explanation

Finite sets have no intervals inside them, so they have no interior points.

6) If int(A) = ∅, then:

A must be countable

A has no open subsets except ∅

A is always finite

A is open

If the interior is empty, A contains no nonempty open interval.

Explanation

If the interior is empty, A contains no nonempty open interval.

7) If int(A) = ∅ but A ≠ ∅, then:

A has no limit points

A is open

A equals its closure

A contains no intervals

Empty interior means A contains no open intervals.

Explanation

Empty interior means A contains no open intervals.

8) The interior of an open interval (a,b) in ℝ is itself.

True

False

Open intervals are already open.

Explanation

Open intervals are already open.

9) A single point in ℝ has no interior points.

True

False

A point cannot contain an open interval around itself.

Explanation

A point cannot contain an open interval around itself.

10) The interior of the union of set X and set Y is always the union of their interiors.

True

False

Interior(X ∪ Y) contains the union of interiors, but may be bigger.

Explanation

Interior(X ∪ Y) contains the union of interiors, but may be bigger.

11) The interior of the intersection of set X and set Y is always the intersection of their interiors.

True

False

Interior distributes over intersection correctly.

Explanation

Interior distributes over intersection correctly.

12) The interior of any subset of ℝ must be either empty or infinite.

True

False

Any nonempty open set in ℝ contains an interval, which has infinitely many points.

Explanation

Any nonempty open set in ℝ contains an interval, which has infinitely many points.

13) Finite closed sets always have a nonempty interior.

True

False

Finite sets have no interior—they contain no intervals.

Explanation

Finite sets have no interior—they contain no intervals.

14) A set can have interior points even if it is not open.

True

False

Example: [0,1] is not open but has interior (0,1).

Explanation

Example: [0,1] is not open but has interior (0,1).

15) A set with empty interior can still contain an open interval.

True

False

If a set had an open interval, that interval would be part of its interior.

Explanation

If a set had an open interval, that interval would be part of its interior.

Interior Points Properties Quiz

One last step: How did you like this quiz?

What do you like, dislike, and how can we improve?