Open Sets Properties Quiz

1) Which of the following is always open in any topology?

Any finite set

Empty set

Any infinite set

None of the above

The empty set is always open in every topology by definition.

Explanation

The empty set is always open in every topology by definition.

2) The intersection of two open sets is:

Always open

Always closed

Sometimes open, sometimes closed

Never open

The intersection of two open sets is always open.

Explanation

The intersection of two open sets is always open.

3) Which of the following is open in ℝ?

{1}

[0,1]

(−∞,5)

{x : x ≥ 2}

(−∞,5) is an open interval. The other sets include boundary points.

Explanation

(−∞,5) is an open interval. The other sets include boundary points.

4) Which of the following is open in ℝ?

[0,1]

[0,1)

(0,1]

(0,1)

(0,1) excludes both endpoints, so it is open.

Explanation

(0,1) excludes both endpoints, so it is open.

5) A set that is both open and closed is called:

Convex

Clopen

Compact

Dense

A “clopen” set is both open and closed.

Explanation

A “clopen” set is both open and closed.

6) The union of (0,1) and (2,3) is:

Open

Closed

Neither

Both

(0,1) and (2,3) are open intervals. Their union is also open.

Explanation

(0,1) and (2,3) are open intervals. Their union is also open.

7) The intersection of (−1,1) and (0,2) is:

Open

Closed

Neither

Both

The intersection is (0,1), which is open.

Explanation

The intersection is (0,1), which is open.

8) The union of (0,1] and (2,3) is:

Open

Closed

Neither

Both

(0,1] is not open, so the union is not open; it's also not closed.

Explanation

(0,1] is not open, so the union is not open; it's also not closed.

9) In ℝ, the set {1/n : n ∈ ℕ} is:

Open

Closed

Neither

Clopen

The set has limit point 0 (not included), so it is not closed, and it's not open. Hence neither.

Explanation

The set has limit point 0 (not included), so it is not closed, and it's not open. Hence neither.

10) Which is an open set in ℝ?

(1,2] ∪ (3,4)

(1,2) ∪ (3,4)

[1,2] ∪ (3,4)

[1,2) ∪ [3,4)

(1,2) and (3,4) are open intervals; their union is open.

Explanation

(1,2) and (3,4) are open intervals; their union is open.

11) Set B = (1,−2] ∪ [3,4) is an open set.

True

False

Both intervals include endpoints, so the set is not open.

Explanation

Both intervals include endpoints, so the set is not open.

12) A set with no interior points cannot be open.

True

False

Open sets must contain at least one interior point.

Explanation

Open sets must contain at least one interior point.

13) Set C = {x ∈ ℝ : x² + 1 < 2} is an open set.

True

False

x² + 1

Explanation

x² + 1

14) If every sequence in set A that converges in the space has its limits inside A, then A must be open.

True

False

This condition describes a closed set, not an open one.

Explanation

This condition describes a closed set, not an open one.

15) The set A = {x ∈ ℝ : x² < 9} is an open set.

True

False

x²

Explanation

x²