Questions: Separability

Question 1

A topological space X is a separable metric space. Which of the following is guaranteed?

• A. X has a countable dense subset, but may or may not have a countable basis for its topology
• B. X is second-countable — it has a countable basis — because separability and second-countability are equivalent in metric spaces
• C. Every subspace of X is also separable, since separability passes to arbitrary subspaces
• D. X must be compact, because separable metric spaces have finite open covers

Answer: (B)

In metric spaces specifically, separability and second-countability are equivalent: every separable metric space is second-countable, and vice versa. This is a special property of metric spaces — in general topological spaces, separability does not imply second-countability. The equivalence is powerful because second-countability enables many compactness and covering arguments, so proving separability in a metric space unlocks these tools. Options A and C contain errors: A gets the metric-space case backwards, and separability does not pass to arbitrary subspaces in general topology (though it does in metric spaces).

Question 2

Which of the following spaces is NOT separable?

• A. ℝ with the standard topology
• B. ℝⁿ with the standard topology
• C. L²([0,1]) — square-integrable functions on [0,1]
• D. The space of all real-valued functions on ℝ with the product topology (ℝ^ℝ)

Answer: (D)

ℝ^ℝ — the space of all real-valued functions on ℝ — is not separable. Its cardinality is 2^{2^ℵ₀}, and no countable set can be dense in it with the product topology. By contrast, ℝ and ℝⁿ are separable (rationals and rational-coordinate points, respectively). L²([0,1]) is separable: polynomials with rational coefficients (or step functions with rational heights and rational endpoints) form a countable dense subset. Separability is not automatic for all 'natural' function spaces — it depends on the topology and the domain.

Question 3

Every second-countable topological space is separable.

• T. True
• F. False

Answer: True

This is a general theorem: if X has a countable basis {B₁, B₂, B₃, …}, pick one point xₙ from each nonempty basis element Bₙ. The resulting countable set {xₙ} is dense: any nonempty open set U contains some basis element Bₙ ⊆ U, and thus contains xₙ. So every second-countable space is separable. The converse holds in metric spaces (separable ⟺ second-countable) but fails in general topology — there exist separable spaces that are not second-countable.

Question 4

In any topological space, separability is equivalent to second-countability — a space is separable if and only if it has a countable basis.

• T. True
• F. False

Answer: False

This equivalence holds in metric spaces but fails in general topology. The 'Sorgenfrey plane' (ℝ² with the lower-limit topology) is a famous counterexample: it is separable (the rationals are dense) but not second-countable (it has no countable basis). The direction 'second-countable ⟹ separable' is always true, but 'separable ⟹ second-countable' requires the metric assumption. Conflating these in general topology is a common error that leads to incorrect proofs.

Question 5

Why is separability described as a 'smallness' or 'tameness' condition on a topological space, and what does it enable analytically?

Model answer: Separability says the space can be approximated by countably many points — there is a countable set dense enough that every open region contains one of them. This 'smallness' is what makes infinite-dimensional spaces tractable: in separable Hilbert spaces, every element can be expressed as a countable series expansion (Fourier series), and limits of such series stay in the space. In metric spaces, separability implies second-countability, which enables compactness arguments, covering theorems, and metrization results. Without separability, spaces can be so large that standard analysis tools (sequences, series, countable covers) fail to capture their structure.

The analogy: ℚ is a 'small' subset of ℝ in cardinality, but it is 'everywhere' in ℝ in the sense of density — every real number can be approximated arbitrarily closely by rationals. Separability generalizes this to arbitrary topological spaces. It is the minimum condition ensuring that approximation by sequences (rather than uncountable nets) can do real analytical work. This is why L^p spaces for p < ∞ are separable — approximation by step functions is effective — while L^∞ is not, and standard Fourier-series methods fail in L^∞.