Questions: Topological Spaces: Definition and Examples

Question 1

Let X = {1, 2, 3}. Which of the following collections τ fails to be a topology on X?

• A. τ = {∅, X}
• B. τ = {∅, {1}, X}
• C. τ = {∅, {1}, {2}, X}
• D. τ = {∅, {1}, {2}, {1,2}, X}

Answer: (C)

Option C fails because it violates the closure under arbitrary unions axiom: {1} and {2} are both in τ, but their union {1,2} is not. A topology must be closed under all unions of its members. Option A (the indiscrete topology) and option B are valid topologies — check that ∅ and X are present, unions stay in τ, and finite intersections stay in τ. Option D includes {1,2} = {1} ∪ {2}, so it is closed under unions and is a valid topology.

Question 2

Topology requires closure under arbitrary unions but only finite intersections. Why is this asymmetry necessary?

• A. It is a historical convention that has no mathematical justification
• B. Infinite intersections are not valid set-theoretic operations
• C. In the standard topology on ℝ, the intersection of all intervals (−1/n, 1/n) is {0}, which is not open — so allowing arbitrary intersections would exclude this standard example
• D. Finite intersections are computationally tractable, while infinite ones are not

Answer: (C)

The axioms are designed to capture the behavior of open sets in metric spaces like ℝ. In the standard topology on ℝ, arbitrary unions of open intervals are open, but the intersection of all open intervals (−1/n, 1/n) for n = 1, 2, 3, ... equals the single point {0}, which is not open. If we required closure under arbitrary intersections, this standard topology would not qualify. The asymmetry is not a convention — it is forced by the intended model.

Question 3

Nearly every set X has exactly one possible topology.

• T. True
• F. False

Answer: False

Any set X carries at least two topologies: the indiscrete topology τ = {∅, X} (only the empty set and X itself are open) and the discrete topology τ = 𝒫(X) (every subset is open). Between these extremes, most sets admit many more topologies. The same set X can be equipped with different topologies, yielding different topological spaces with different notions of openness, continuity, and convergence. The topology is not determined by the set — it is an additional structure you impose.

Question 4

In any topological space (X, τ), both the empty set ∅ and the whole set X must be declared open.

• T. True
• F. False

Answer: True

This is the first axiom of a topology: ∅ ∈ τ and X ∈ τ. Both inclusions serve important technical roles. ∅ being open is required for consistency with the union and intersection axioms (the empty union is ∅, which must be in τ). X being open ensures the whole space is 'open,' which is needed for many topological arguments. These are not optional — any collection of subsets that fails to include both ∅ and X is not a topology by definition.

Question 5

Why does defining 'openness' through axioms rather than through distance allow topology to generalize beyond metric spaces?

Model answer: Distance imposes a rigid quantitative structure: two points are 'close' if a number (their distance) is small. The topological axioms abstract only the qualitative behavior of open sets — which sets can be unioned and intersected — without requiring any measurement. This means the framework applies to any collection of subsets satisfying the axioms, including spaces with no natural notion of distance: function spaces, abstract sets with only combinatorial structure, spaces of convergent sequences, and more. Topology studies what is preserved by continuous deformation, not by rigid motion, and distance is too strong a requirement for that.

The key move is identifying which properties of open sets in ℝ are actually used in proofs about continuity and convergence — and it turns out to be only the three axioms, not the full distance structure. Once the axioms are in place, you can define continuity (preimages of open sets are open), convergence, compactness, and connectedness purely in terms of the topology τ, with no reference to distance. Metric spaces become a special case of topological spaces, and all theorems proved topologically apply to them automatically.