In ℝ with the standard topology, the interval [0, 1] is closed. What is the correct reason, based on the definition of a closed set?
AIt contains both of its endpoints
BIts complement (−∞, 0) ∪ (1, ∞) is an open set
CIt is bounded and therefore closed
DIt is compact, and compact sets are always closed
The definition of a closed set is: F is closed if X \ F is open. The complement of [0,1] in ℝ is (−∞, 0) ∪ (1, ∞), which is a union of open intervals and therefore open. This is why [0,1] is closed — not because it contains its endpoints (that is a consequence, not a definition) and not because it is bounded or compact (which require additional theory).
Question 2 Multiple Choice
Consider an infinite collection of closed sets F₁, F₂, F₃, ... in ℝ. Let A = ∩ Fₙ (intersection of all) and B = ∪ Fₙ (union of all). Which is guaranteed to be closed?
AA but not necessarily B
BB but not necessarily A
CBoth A and B
DNeither A nor B
Arbitrary intersections of closed sets are always closed (by De Morgan's law: the complement of an arbitrary intersection is an arbitrary union of open sets, which is open). Arbitrary unions of closed sets need not be closed. Counterexample: Fₙ = [1/n, 1] for n = 1, 2, 3, ... Each Fₙ is closed, but their union ∪ Fₙ = (0, 1], which is not closed because its complement (−∞, 0] ∪ (1, ∞) is not open.
Question 3 True / False
The empty set ∅ is both open and closed in any topological space.
TTrue
FFalse
Answer: True
The axioms of a topology require that ∅ and X are both open. Since ∅ is open, its complement X must also be open (X is required to be open by the axioms). Therefore ∅ = X \ X has an open complement, making ∅ closed. Similarly X is both open and closed. Sets that are both open and closed are called 'clopen.'
Question 4 True / False
Nearly every subset of a topological space is either open, closed, or both.
TTrue
FFalse
Answer: False
A set can be neither open nor closed. In ℝ with the standard topology, the interval (0, 1] is an example: it is not open (the point 1 has no open ball around it contained in (0,1]) and not closed (its complement (−∞, 0] ∪ (1, ∞) is not open, since no open ball around 0 is contained in that set). The four possibilities — open only, closed only, both, neither — all genuinely occur.
Question 5 Short Answer
Why is it incorrect to *define* a closed set as 'a set that contains all its boundary points,' even though this is sometimes stated as a characterization of closed sets in metric spaces?
Think about your answer, then reveal below.
Model answer: The actual definition of a closed set is that its complement is open — this is what the axioms give us. In metric spaces, it can be proved that this is equivalent to containing all limit points (and hence all boundary points), but that equivalence is a theorem, not a definition. In a general topological space, 'boundary point' and 'limit point' are themselves defined in terms of open and closed sets, so using them to define closed sets would be circular. The complement-is-open definition is the foundational one.
This distinction matters when moving beyond metric spaces. In a general topology, intuitive notions like 'boundary' and 'limit point' don't automatically apply, so the clean set-theoretic definition (complement is open) is what remains universally valid.