Questions: Open Sets in Topological Spaces

Question 1

In the discrete topology on a non-empty set X (where τ = P(X), the power set), which statement is true?

• A. Only X and ∅ are open
• B. Every subset of X is open
• C. Only finite subsets are open
• D. A set is open if and only if it contains at least one point

Answer: (B)

By definition, the discrete topology on X takes τ = P(X) — every possible subset is declared open. This is the finest possible topology on X. In contrast, the indiscrete topology takes τ = {∅, X}, making option A true for that topology. Options C and D do not correspond to any standard topology.

Question 2

In any topological space, a set can be simultaneously open and closed.

• T. True
• F. False

Answer: True

Such sets are called 'clopen.' In every topological space, ∅ and X are both open (by the axioms) and closed (each is the complement of the other open set). In the discrete topology, every set is clopen. A connected space is precisely one where the only clopen sets are ∅ and X, so clopen sets are a meaningful concept, not a contradiction.

Question 3

Why can't we define open sets in a general topological space by saying 'a set is open if every point has a positive distance to the boundary'?

Model answer: A general topological space may have no notion of distance at all. Open sets are defined purely by membership in the collection τ satisfying three axioms (containing ∅ and X, closed under arbitrary unions, closed under finite intersections), which requires no metric.

The distance-to-boundary characterization works in metric spaces (like ℝⁿ) where d(x, boundary) > 0 captures open sets perfectly. But topology was developed precisely to study spaces where distance is absent or irrelevant — function spaces, quotient spaces, abstract manifolds. The axiomatic definition via τ captures the essential 'neighborhood' structure without needing a metric.