In the standard topology on ℝ, can a set be both open and closed at the same time?
ANo — a set is either open or closed, never both, because the definitions are mutually exclusive
BYes, but only the empty set has this property
CYes — the empty set and ℝ itself are both open and closed in any topology
DYes, but only in discrete topologies, not standard ones
In any topological space, both the empty set and the whole space are simultaneously open and closed — they are called 'clopen' sets. The empty set's complement is the whole space (open), so it is closed; the whole space's complement is the empty set (open), so it too is closed. The idea that open and closed are mutually exclusive is a common misconception — the definitions are not opposites but are defined independently via the complement relationship.
Question 2 Multiple Choice
In the standard topology on ℝ, which of the following sets is closed?
A(0, 1) — because it is a bounded interval
B[0, 1] — because its complement (−∞, 0) ∪ (1, ∞) is open
C(0, ∞) — because it extends to infinity
D[0, 1) — because it contains its left endpoint
A set is closed if and only if its complement is open. The complement of [0,1] is (−∞,0) ∪ (1,∞), which is a union of open intervals and therefore open — so [0,1] is closed. The set (0,1) is open (not closed), [0,1) contains 0 but its complement (−∞,0) ∪ [1,∞) is not open (since 1 is a boundary point with no open neighborhood contained in the complement), and (0,∞) is open.
Question 3 True / False
The union of infinitely many closed sets is typically closed.
TTrue
FFalse
Answer: False
Closed sets are closed under finite unions, but NOT arbitrary (infinite) unions. A classic counterexample: the sets {1/n} for n = 1, 2, 3, … are each closed in ℝ (single points are closed), but their union {1, 1/2, 1/3, …} is not closed — the limit point 0 is not in the set, yet every neighborhood of 0 intersects the union. Contrast with intersections: arbitrary intersections of closed sets ARE always closed.
Question 4 True / False
In the standard topology on ℝ, the set [0, 1] is closed because its complement (−∞, 0) ∪ (1, ∞) is open.
TTrue
FFalse
Answer: True
This is the direct application of the definition: a set C is closed if and only if its complement is open. The complement of [0,1] in ℝ is (−∞,0) ∪ (1,∞), which is a union of two open intervals and therefore open in the standard topology. Hence [0,1] is closed. This also illustrates why [0,1] 'includes its boundary points' — removing either endpoint would leave a set whose complement is no longer open.
Question 5 Short Answer
Why is a set defined as 'closed' when its complement is open, rather than closed sets being defined by their own direct properties?
Think about your answer, then reveal below.
Model answer: In topology, 'open sets' are the primitive objects — they are specified directly by the axioms of a topology. Closed sets are then defined derivatively as complements of open sets, which allows the entire theory of closed sets (and their properties like containing limit points) to be derived from the open set axioms without introducing new axioms. This duality means that every theorem about open sets has a dual theorem about closed sets, making the framework economical.
The definition-by-complement approach reflects the foundational structure of topology: the open sets are the basic data from which everything else is built. Defining closed sets as complements of open sets is not circular — it's a deliberate design choice that makes the open/closed duality a theorem (not an axiom) and ensures that the familiar properties of closed sets (containing limit points, being stable under arbitrary intersection) follow automatically from the open set axioms.