Consider the set S = {1/n : n ∈ ℕ} = {1, 1/2, 1/3, 1/4, …}. Is S a closed subset of ℝ?
AYes, because all elements of S are positive, so S is bounded away from negative numbers
BYes, because S is bounded above by 1 and below by 0, satisfying the definition of a closed set
CNo, because the sequence 1/n converges to 0, which is a limit point of S not contained in S
DNo, because S contains infinitely many points, and infinite sets cannot be closed
A closed set must contain all its limit points. The sequence 1, 1/2, 1/3, … converges to 0, making 0 a limit point of S. But 0 ∉ S, so S fails the closed set criterion. Adding 0 — forming {0} ∪ {1/n : n ≥ 1} — makes the set closed. Options A and B describe properties (positivity, boundedness) that are irrelevant to the definition of closedness.
Question 2 Multiple Choice
Which statement correctly describes the duality between open and closed sets with respect to unions and intersections?
AArbitrary unions of closed sets are closed; only finite intersections of open sets are open
BOnly finite unions of closed sets are guaranteed to be closed; arbitrary intersections of closed sets are closed
CBoth open sets and closed sets are closed under arbitrary unions
DClosed sets are closed under countable unions but not arbitrary ones
The exact dual of the open-set rule (arbitrary unions open; finite intersections open) is: arbitrary intersections closed; finite unions closed. Classic counterexample for infinite unions of closed sets: each singleton {1/n} is closed (finite sets are closed), but ⋃{1/n} = {1, 1/2, 1/3, …} is not closed, as shown above.
Question 3 True / False
The interval [0, 1) is a closed set because it contains the boundary point 0.
TTrue
FFalse
Answer: False
[0, 1) is neither open nor closed. The sequence of points 1 − 1/n = {0, 1/2, 2/3, 3/4, …} lies entirely inside [0, 1) and converges to 1. But 1 ∉ [0, 1), so 1 is a limit point not in the set — [0, 1) fails the limit-point criterion. The fact that 0 is included does not make the set closed; what matters is whether *all* limit points are included.
Question 4 True / False
A set that is not open should be closed.
TTrue
FFalse
Answer: False
This is a very common misconception. 'Open' and 'closed' are independent properties — they are not logical opposites. A set can be: open but not closed (e.g., (0,1)); closed but not open (e.g., [0,1]); both open and closed (e.g., ℝ itself, or ∅); or neither open nor closed (e.g., [0,1)). The interval [0,1) is not open (0 has no neighborhood entirely inside it) and not closed (1 is a limit point not in the set).
Question 5 Short Answer
What does it mean for a set to 'contain all its limit points,' and why does this property capture the intuition of being 'closed'?
Think about your answer, then reveal below.
Model answer: A limit point of F is any point x such that every neighborhood of x contains a point of F other than x — sequences from F can get arbitrarily close to x. F is closed if every such limit point belongs to F. This captures 'closedness' because it means F is stable under the most basic operation of analysis: taking limits. You cannot escape F by taking limits of sequences from F. An open set like (0,1) fails because sequences approaching 0 or 1 from inside converge to points outside the set.
The limit-point criterion makes closed sets the natural setting for analysis: whenever you have a sequence from a closed set and it converges, the limit is guaranteed to stay in the set. This is why closed and bounded (compact) sets are so powerful — you can extract convergent subsequences without worrying about limits escaping.