Basics Of Open Sets Quiz

1) Every open interval in the real numbers, such as (2,5), is an open set in the standard topology on ℝ.

True

False

True, an open interval like (2,5) does not include its endpoints, so every point has a small interval around it that stays inside, which makes it open.

Explanation

True, an open interval like (2,5) does not include its endpoints, so every point has a small interval around it that stays inside, which makes it open.

2) The union of any collection of open sets is always an open set.

True

False

If a point is in the union, it belongs to at least one open set, so it has a neighborhood inside that set and therefore the union is open.

Explanation

If a point is in the union, it belongs to at least one open set, so it has a neighborhood inside that set and therefore the union is open.

3) The intersection of two open sets is always an open set.

True

False

A point in both open sets has a small neighborhood inside each one; a smaller neighborhood stays in the intersection, so the intersection is open.

Explanation

A point in both open sets has a small neighborhood inside each one; a smaller neighborhood stays in the intersection, so the intersection is open.

4) The empty set ∅ is considered an open set.

True

False

By definition in topology, the empty set is always an open set.

Explanation

By definition in topology, the empty set is always an open set.

5) An interval [1,3] is considered an open set.

True

False

[1,3] includes its endpoints, so you cannot place a small interval around 1 or 3 that stays inside the set; therefore it is not open.

Explanation

[1,3] includes its endpoints, so you cannot place a small interval around 1 or 3 that stays inside the set; therefore it is not open.

6) A set is open if it does not contain its boundary points.

True

False

In ℝ, open sets exclude all of their boundary points; that is a basic characterization of openness.

Explanation

In ℝ, open sets exclude all of their boundary points; that is a basic characterization of openness.

7) The complement of a closed set is an open set.

True

False

In topology, a set is closed if and only if its complement is open, so the complement of a closed set is open.

Explanation

In topology, a set is closed if and only if its complement is open, so the complement of a closed set is open.

8) A set that is not closed must be open.

True

False

A set can be neither open nor closed; for example, [0,1) in ℝ is not open and not closed.

Explanation

A set can be neither open nor closed; for example, [0,1) in ℝ is not open and not closed.

9) If all the boundary points are included in the set, then it is an open set.

True

False

If all boundary points are included, the set is closed, not open.

Explanation

If all boundary points are included, the set is closed, not open.

10) Set A = {x : −3 < x < 3, x ∈ ℝ} is an open set.

True

False

This is the open interval (−3,3), which does not include its endpoints, so it is an open set.

Explanation

This is the open interval (−3,3), which does not include its endpoints, so it is an open set.

11) Which of the following can be both open and closed?

Any set with interior

Any bounded set

Any infinite set

The empty set

The empty set is one of the special sets (along with the whole space) that is both open and closed (clopen).

Explanation

The empty set is one of the special sets (along with the whole space) that is both open and closed (clopen).

12) Which of the following statements about open sets is false?

Arbitrary unions of open sets are open

Finite intersections of open sets are open

The entire space is open

Countable intersections of open sets are always open

Countable intersections of open sets need not be open; for example, the intersection of (−1/n, 1/n) over n gives {0}, which is not open in ℝ.

Explanation

Countable intersections of open sets need not be open; for example, the intersection of (−1/n, 1/n) over n gives {0}, which is not open in ℝ.

13) A set U in a topological space (X,τ) is called open if:

Its complement is closed

Every point in U is an interior point

U contains all limit points of x

U is finite

A set is open if every point in it is an interior point, meaning each point has a neighborhood contained entirely in U.

Explanation

A set is open if every point in it is an interior point, meaning each point has a neighborhood contained entirely in U.

14) In ℝ², the set {(x,y): x² + y² < 1} is:

Open

Closed

Clopen

Neither

This is the interior of the unit disk; it does not include the boundary circle x² + y² = 1, so it is an open set.

Explanation

This is the interior of the unit disk; it does not include the boundary circle x² + y² = 1, so it is an open set.

15) If U is open in ℝ under the standard topology, then:

U can be expressed as a union of open intervals

U must be bounded

U is always finite

U contains all integers

Every open set in ℝ can be written as a union (possibly infinite) of open intervals.

Explanation

Every open set in ℝ can be written as a union (possibly infinite) of open intervals.

Basics of Open Sets Quiz