The rationals ℚ have Lebesgue measure zero in ℝ — they are negligible from a measure-theoretic standpoint. What does this imply about whether ℚ is nowhere dense in ℝ?
AIt implies ℚ is nowhere dense, because a measure-zero set cannot be topologically significant
BNothing — ℚ is actually dense in ℝ, showing that topological density and measure-theoretic size are independent
CIt implies ℚ is nowhere dense, because int(cl(ℚ)) = ∅ follows from measure zero
DIt implies ℚ is neither dense nor nowhere dense — it occupies a middle category
ℚ is dense in ℝ: between any two real numbers lies a rational, so every open interval contains points of ℚ, meaning cl(ℚ) = ℝ. This is the exact opposite of nowhere dense. The critical lesson is that topological density and measure-theoretic size are entirely independent notions — a countable, measure-zero set can be topologically everywhere present. The common misconception is that 'small' (in measure) implies topologically thin. The Cantor set shows the reverse is also possible: uncountable yet nowhere dense.
Question 2 Multiple Choice
A set A in a metric space X is nowhere dense. Which of the following must be true?
AA has measure zero
BA is countable
CThe interior of the closure of A is empty: int(cl(A)) = ∅
DA is closed and contains no limit points
The definition of nowhere dense is precisely int(cl(A)) = ∅ — the closure of A contains no open set. This says nothing about cardinality or measure: the Cantor set is uncountable and nowhere dense; ℤ is countable and nowhere dense. Option A fails: a closed nowhere dense set can have positive measure in principle (though not in ℝ for standard examples). Option D is wrong because the closure can certainly have limit points — it just cannot contain any open interval.
Question 3 True / False
The Cantor set is nowhere dense in ℝ, which implies it should be countable.
TTrue
FFalse
Answer: False
The Cantor set is uncountable — it has the same cardinality as ℝ — yet it is nowhere dense. Its closure is itself (it is closed), and it contains no open interval (every interval removed from [0,1] by the Cantor construction leaves a gap in C). Nowhere dense is a topological notion of 'thinness' that is entirely independent of cardinality. This is why the Baire category framework uses 'meager' rather than 'countable' — they capture different kinds of smallness.
Question 4 True / False
A set D is dense in a metric space X if and only if every non-empty open set in X contains at least one point of D.
TTrue
FFalse
Answer: True
This is equivalent to the definition cl(D) = X. If every open set contains a point of D, then every point of X is either in D or is a limit point of D, so cl(D) = X. Conversely, if cl(D) = X and U is a non-empty open set, any point x ∈ U is either in D (done) or a limit point of D, so every neighborhood of x — including U — meets D. This characterization makes dense sets operationally useful: to check density, check that no open set 'misses' the set.
Question 5 Short Answer
What is the significance of the Baire Category Theorem, and what does it say about complete metric spaces and nowhere dense sets?
Think about your answer, then reveal below.
Model answer: The Baire Category Theorem states that a complete metric space cannot be expressed as a countable union of nowhere dense sets — it is non-meager. This means that the 'typical' or 'generic' element of the space cannot be excluded by any finite or countable collection of topologically thin sets. It is used in analysis to prove existence results without constructing explicit examples, such as showing that most continuous functions are nowhere differentiable.
The theorem's power lies in what it rules out: in a complete space, you cannot cover everything with countably many 'thin' (nowhere dense) pieces. This is why it yields strong generic statements — it shows that the 'complement' of a meager set is topologically large, meaning a typical element of the space has some desired property. The contrast with measure theory is instructive: a meager set can have full measure, and a measure-zero set can be non-meager (like ℚ), underscoring that Baire category and Lebesgue measure capture orthogonal aspects of 'size.'