Consider the set X = {1, 2, 3} with topology τ = {∅, {1}, {1,2}, X}. Is {2} an open set in this topology?
AYes — every singleton set is open in any topology
BNo — {2} is not in τ, and open sets are exactly the members of τ
CYes — {2} is a subset of {1,2}, which is open, so {2} must be open
DNo — {2} is not open because it doesn't contain 1, the smallest element
In a topological space (X, τ), a set is open if and only if it belongs to the topology τ. Here τ = {∅, {1}, {1,2}, X}, and {2} is not listed — so {2} is not open. Option A is wrong: singletons are not guaranteed to be open in a general topology (they are open in the discrete topology but not in others). Option C confuses 'subset of an open set' with 'open' — this would make every subset of X open, which is only true for the discrete topology.
Question 2 Multiple Choice
Which of the following is guaranteed to be open in every possible topology on a set X?
AEvery singleton set {x} for each x in X
BOnly ∅ and X itself
CEvery finite subset of X
DEvery subset of X — all subsets are open by default
The topology axioms require that both ∅ and X itself belong to any topology — these are the only sets that must be open in every topology. Everything else depends on the choice of topology. The discrete topology makes every subset open; the indiscrete topology makes only ∅ and X open. Singletons are not always open (they fail in the indiscrete topology). Finite subsets are not always open (they may be absent from a topology). 'Openness' is a property conferred by the topology, not intrinsic to the set.
Question 3 True / False
In a topological space, an arbitrary (possibly infinite) intersection of open sets is generally open.
TTrue
FFalse
Answer: False
The open-set axioms only guarantee that finite intersections of open sets are open. Infinite intersections can fail to be open. A classic example in ℝ with the standard topology: the open intervals (−1/n, 1/n) are open for every positive integer n, but their intersection ⋂ₙ (−1/n, 1/n) = {0}, a single point, which is not open in ℝ. The restriction to finite intersections is not an oversight — it is carefully designed to exclude this failure while preserving enough structure for the theory to work.
Question 4 True / False
The same set of points S can be an open set in one topology on X but fail to be open in a different topology on the same set X.
TTrue
FFalse
Answer: True
Openness is relative to a topology, not intrinsic to the set. For example, take X = {1, 2, 3}. In the discrete topology (where every subset is open), {1} is open. In the indiscrete topology τ = {∅, X}, only ∅ and X are open — so {1} is not open. The 'same' set {1} is open in one topology and not open in another. This is a key conceptual shift from metric spaces, where openness is determined by distances and is less flexible.
Question 5 Short Answer
Why does the topology axiom require only finite intersections of open sets to be open, rather than allowing arbitrary intersections? What goes wrong if you permit infinite intersections?
Think about your answer, then reveal below.
Model answer: Infinite intersections of open sets can be closed sets — or more precisely, sets that are not open. In the standard topology on ℝ, the intersection of all open intervals (−1/n, 1/n) for n = 1, 2, 3, … is the single point {0}, which is a closed set (not open). If infinite intersections were required to be open, then {0} would be open, but then every set would be open (by taking intersections of intervals centered at any point), collapsing the topology to the discrete topology. The finite-intersection axiom is precisely calibrated to preserve a non-trivial distinction between open and non-open sets.
This is why topology textbooks are careful to state 'finite intersection' explicitly. The restriction is load-bearing: it separates the useful structure of topological spaces from the too-rigid discrete topology where everything is open and all maps are automatically continuous.