Questions: Open and Closed Sets on the Real Line

Question 1

The half-open interval [0, 1) — containing 0 but not 1 — is best classified as:

• A. Open, because most of its points have breathing room
• B. Closed, because it contains its left endpoint 0
• C. Neither open nor closed
• D. Both open and closed

Answer: (C)

[0, 1) is neither open nor closed. It fails to be open because the point 0 has no open interval around it that stays inside [0, 1) — any interval around 0 reaches negative numbers. It fails to be closed because 1 is a limit point of the set (sequences like 1 − 1/n converge to 1) but 1 is not in the set. 'Closed' does not simply mean 'has a closed endpoint' — it means the set contains all its limit points.

Question 2

In a proof, you show that a sequence (x_n) in a set F converges to a limit L. Which condition on F guarantees that L ∈ F?

• A. F is open
• B. F is closed
• C. F is bounded
• D. F is a subset of ℝ

Answer: (B)

A set is closed if and only if it contains all its limit points — equivalently, if every convergent sequence in F has its limit in F. Openness says nothing about limits; it says every point has a neighborhood inside F. Boundedness and being a subset of ℝ are not sufficient. Closed sets are precisely the sets 'stable under limits,' which is why analysis theorems conclude 'the limit lies in F' when F is closed.

Question 3

The empty set ∅ and the entire real line ℝ are both open and closed simultaneously.

• T. True
• F. False

Answer: True

Both ∅ and ℝ are 'clopen' — they satisfy the definitions of both open and closed. ∅ is open vacuously (there are no points in it that fail to have a neighborhood inside it) and closed vacuously (there are no limit points it could fail to contain). ℝ is open because every point has an open interval inside ℝ, and closed because its complement ∅ is open. This shows that open and closed are not logical opposites: a set can be both, one, or neither.

Question 4

Nearly every subset of ℝ is expected to be either open or closed — there is no middle ground.

• T. True
• F. False

Answer: False

This is false. The half-open interval [0, 1) is a standard example of a set that is neither open nor closed. It is not open because the boundary point 0 has no open neighborhood contained in [0, 1). It is not closed because 1 is a limit point (approached by sequences in the set) but 1 ∉ [0, 1). The open/closed distinction is not a partition of all subsets — 'neither' is a genuine fourth category alongside 'open only,' 'closed only,' and 'both.'

Question 5

Explain why a set being 'closed' in the mathematical sense does not mean the same thing as 'not open,' and give an example illustrating the difference.

Model answer: A closed set is one that contains all its limit points (or equivalently, whose complement is open). 'Not open' simply means some point lacks a neighborhood inside the set. A closed set can also be open — ℝ itself is both. A set can be neither — [0, 1) fails both definitions. The everyday opposites 'open/closed' do not apply here: the two properties are logically independent, defined by different conditions (limit points vs. interior neighborhoods), and can co-occur or both fail.

The key conceptual trap is importing everyday language into mathematics. In topology, open and closed are independent properties defined formally. Recognizing this prevents errors like concluding 'if F is not open, it must be closed' — a common false inference that breaks down for sets like [0, 1) or any half-open interval.