Questions: Topological Spaces: Definition and Examples

Question 1

Let X = {a, b, c} and τ = {∅, {a}, {a, b}, X}. Is the set {b} open in the topology τ?

• A. Yes — {b} is a subset of X, and all subsets of X are open in any topology
• B. No — {b} is not in τ, and openness means being a member of the topology
• C. Yes — {b} is a subset of the open set {a, b}, so it must be open
• D. The question cannot be answered without knowing the metric on X

Answer: (B)

In a topological space, 'open' means precisely 'is a member of τ.' The set {b} is not listed in τ = {∅, {a}, {a,b}, X}, so it is not open. Option A describes the discrete topology, which τ is not. Option C confuses 'subset of an open set' with 'open' — subsets of open sets are not automatically open. Option D reflects the misconception that topologies must come from metrics; τ is a topology by definition, and no metric is needed or referenced.

Question 2

The indiscrete topology on X = {a, b} has τ = {∅, X}. A student claims that {a} is open because 'a is a point in X and points should be open.' What is wrong with this reasoning?

• A. The student is correct — points are always open in any topology
• B. The student conflates 'open' with 'nonempty' — open is a topological designation, not a property of single elements
• C. 'Open' is not intrinsic — it depends entirely on which sets belong to τ, and {a} ∉ τ in the indiscrete topology
• D. The student should check whether {a} is also closed before deciding

Answer: (C)

The student is importing an intuition from the discrete topology (where every singleton is open) and incorrectly treating it as universal. 'Open' is entirely determined by membership in τ — it is a declaration about the topological structure, not a property a set possesses independently. In the indiscrete topology, only ∅ and X are open. {a} is neither. This is the most important conceptual shift from calculus to topology: 'open' is not intrinsic to a set; it is relative to a chosen topology on the containing space.

Question 3

A set that is open in the discrete topology on X may not be open if a different topology is placed on the same set X.

• T. True
• F. False

Answer: True

This is the central conceptual point: 'open' is relative to the topology, not intrinsic to the set. In the discrete topology on X = {a, b, c}, every subset — including {a}, {b}, {a,c}, etc. — is open. But in the indiscrete topology on the same X, only ∅ and X are open, and {a} is not open. Same set X, same subset {a}, different topologies — different answers to 'is {a} open?' This dependence is what makes topology a study of structure rather than just a study of sets.

Question 4

Most topology on a set X is expected to come from a metric (a distance function) on X — a topology is essentially the same thing as a metric space.

• T. True
• F. False

Answer: False

This is the most common misconception about topology. A topological space is strictly more general than a metric space. The indiscrete topology on a set with two or more points is not metrizable — no metric can generate it, because metric spaces always produce the T₁ property (singletons are closed), but the indiscrete topology is not T₁. One of topology's founding motivations was precisely to identify which properties of analysis require a metric and which require only the open-set axioms — many deep theorems hold in all topological spaces, not just metric ones.

Question 5

The axiom for topological spaces allows arbitrary unions but only finite intersections of open sets. Give an example on ℝ (with the standard topology) showing why infinite intersections of open sets can fail to be open.

Model answer: For each n ≥ 1, the interval (−1/n, 1/n) is open in ℝ. But the intersection ⋂_{n=1}^∞ (−1/n, 1/n) = {0}, the singleton containing only 0. In the standard topology on ℝ, no singleton is open. So infinitely many open sets can intersect to produce a non-open set. This is why the topology axiom requires only finite intersections to be open — finite intersections of open neighborhoods remain open (needed for continuity), but infinite intersections can collapse to a point.

The asymmetry between arbitrary unions and only finite intersections is deliberate. Taking bigger and bigger unions of open sets keeps you within open sets (you're only widening). But intersecting open intervals can progressively shrink them to a single point, which should not be forced to be open — that would make the standard topology on ℝ collapse to the discrete topology. The axioms are calibrated to capture exactly the structure needed for continuity while remaining general enough to allow interesting non-metric examples.