Questions: Topological Spaces: Definition and Examples

Question 1

Why does the definition of a topology require finite intersections of open sets to be open, but allows ARBITRARY (including infinite) unions of open sets to be open?

• A. Finite intersections are computationally easier to verify, so the axiom is a practical convenience
• B. Infinite unions can always be reduced to finite ones by compactness, so infinite unions are not a meaningful additional requirement
• C. Infinite intersections of open sets can fail to be open — for example, the intersection of all intervals (−1/n, 1/n) in ℝ is {0}, which is not open — so allowing them would collapse the structure
• D. The axiom is asymmetric for historical reasons only; both could have been stated for finite collections

Answer: (C)

The asymmetry is mathematically essential, not accidental. In ℝ, the open sets (−1/n, 1/n) for n = 1, 2, 3, … are each open, but their infinite intersection is exactly the single point {0}, which is closed (not open) in the standard topology. Allowing infinite intersections in the axioms would let you build closed sets from open ones through the back door, destroying the distinction between open and closed that topology depends on. Infinite unions pose no such problem: any union of open sets in ℝ is open.

Question 2

On the set X = {a, b, c}, which of the following collections is a valid topology?

• A. τ = {∅, {a}, {b}, X} — contains the required sets and several singletons
• B. τ = {∅, {a}, {a, b}, X} — contains ∅ and X, and is closed under union and finite intersection
• C. τ = {∅, {a}, {b, c}} — contains ∅ and a partition of X into two parts
• D. τ = {{a}, {b}, {c}, X} — contains all singletons and the full set

Answer: (B)

Check τ = {∅, {a}, {a,b}, X}: unions — {a} ∪ {a,b} = {a,b} ✓; {a} ∪ ∅ = {a} ✓; all others stay in τ. Intersections — {a} ∩ {a,b} = {a} ✓; all others ✓. Option A fails: {a} ∪ {b} = {a,b} ∉ τ. Option C fails: it doesn't contain X. Option D fails: it doesn't contain ∅, and {a} ∪ {b} = {a,b} ∉ τ. The key check is always closure under union and finite intersection.

Question 3

In the discrete topology on any set X, every subset of X is an open set.

• T. True
• F. False

Answer: True

The discrete topology is τ = 𝒫(X) — the power set, all subsets of X. It satisfies all three axioms: ∅ and X are in 𝒫(X); any union of subsets of X is a subset of X; any finite intersection of subsets of X is a subset of X. It is the finest possible topology on X — no topology can have more open sets. In a metric space analogue, the discrete topology corresponds to every point being isolated, with every set of points being a union of isolated points and therefore open.

Question 4

Any collection τ of subsets of X that contains both ∅ and X is a valid topology on X.

• T. True
• F. False

Answer: False

Containing ∅ and X is necessary but not sufficient. τ must also be closed under arbitrary unions (any union of sets in τ must be in τ) and closed under finite intersections (any finite intersection of sets in τ must be in τ). For example, on X = {a,b,c}, the collection τ = {∅, {a}, {b}, X} contains ∅ and X but fails closure under union: {a} ∪ {b} = {a,b} ∉ τ. So τ is not a topology despite satisfying the first axiom.

Question 5

What is the key conceptual advantage of defining topology without reference to distance? Why does this generalization matter for mathematics?

Model answer: By replacing the distance-based definition of 'open set' with three axiomatic properties, topology makes the key theorems of analysis (continuity, convergence, connectedness, compactness) apply to any set where those axioms hold — regardless of whether distances make sense. This means a single proof in the topological setting yields results for real analysis, complex analysis, function spaces, manifolds, and purely combinatorial or algebraic structures simultaneously.

The metric definition of open set is specific to spaces where distance is defined. Many important mathematical objects — spaces of functions, quotient spaces, abstract algebraic structures — have meaningful notions of 'nearness' without a natural distance. Topology abstracts the essential behavior (what open sets do) from the specific mechanism (distance). Every theorem proved in pure topological language is automatically a theorem in every metric space, every manifold, and every other topological space. This is the payoff of the axiomatization.