The open interval (0, 1) with the usual metric is homeomorphic to ℝ. What does this imply about the completeness of (0, 1)?
ASince (0, 1) and ℝ are homeomorphic, (0, 1) must also be complete
BSince (0, 1) and ℝ are homeomorphic, they share all metric properties including completeness
CHomeomorphism preserves topological structure, not metric structure — (0, 1) can be incomplete even though ℝ is complete
DThe completeness of ℝ is inherited by any subset of ℝ under the induced metric
Completeness is a metric property, not a purely topological one. Two spaces can be homeomorphic (topologically identical) while differing in completeness. The interval (0, 1) is homeomorphic to ℝ, but (0, 1) is incomplete under the usual metric: the sequence 1/n is Cauchy but converges to 0, which lies outside (0, 1). The real line ℝ is complete. Options A and B embody the common error of conflating metric and topological structure. Option D is false — closed subsets of complete spaces are complete, but (0, 1) is not closed in ℝ.
Question 2 Multiple Choice
Consider the sequence 1, 1.4, 1.41, 1.414, 1.4142, ... of rational decimal approximations to √2. In the metric space (ℚ, |·|), which of the following is true?
AThe sequence is not Cauchy, because its terms never stabilize at a rational number
BThe sequence is Cauchy and converges in ℚ to the irrational number √2
CThe sequence is Cauchy, but it does not converge in ℚ because √2 ∉ ℚ
DThe sequence is Cauchy in ℝ but not in ℚ, since Cauchy-ness depends on the ambient space
The sequence is Cauchy in ℚ: for any ε > 0, successive terms eventually differ by less than ε (the differences go to zero). Being Cauchy depends only on whether the terms become close to each other — it does not require knowing what the limit is. However, the limit √2 is irrational, so it is not in ℚ. The sequence is Cauchy but has no limit within ℚ. This is precisely the failure of completeness: ℚ has 'holes' at irrational numbers that Cauchy sequences can fall into. Option D is wrong because Cauchy-ness uses the same distances whether you consider the sequence in ℚ or in ℝ.
Question 3 True / False
Every convergent sequence in a metric space is a Cauchy sequence, but not every Cauchy sequence converges.
TTrue
FFalse
Answer: True
If a sequence converges to a limit L, then for any ε > 0, terms eventually get within ε/2 of L, so they get within ε of each other — the sequence is Cauchy. The converse fails in incomplete spaces: a Cauchy sequence in ℚ may converge to an irrational number not in ℚ. In a complete metric space, the two notions coincide — every Cauchy sequence converges. This equivalence is exactly what completeness guarantees.
Question 4 True / False
If two metric spaces are homeomorphic, they should have the same completeness.
TTrue
FFalse
Answer: False
Homeomorphism preserves topological structure (open sets, continuity, connectedness) but not metric structure. Completeness is a metric property that depends on the specific distance function, not just the topology. The spaces (0, 1) and ℝ are homeomorphic via a continuous bijection like x → tan(π(x − 1/2)), but ℝ is complete while (0, 1) is not. You can 'change' completeness by picking a different metric or by working in a subspace — something impossible with purely topological properties.
Question 5 Short Answer
What does it mean for a metric space to be 'complete,' and why is the rational number line ℚ a standard example of an incomplete space?
Think about your answer, then reveal below.
Model answer: A metric space is complete if every Cauchy sequence in the space converges to a point that is in the space. ℚ is incomplete because it contains Cauchy sequences whose limits are irrational. For example, the sequence of rational approximations to √2 is Cauchy (terms get arbitrarily close to each other) but has no limit within ℚ, since √2 ∉ ℚ. ℚ has 'holes' at irrational numbers — these are exactly the missing limits that Cauchy sequences try to reach but cannot find.
The definition separates two things: (1) internal coherence of the sequence (Cauchy property — terms cluster together) and (2) existence of a limit in the space. A complete space guarantees that if terms are clustering, there is a point in the space they are clustering toward. The reals ℝ are constructed precisely to fill these holes in ℚ, making ℝ the completion of ℚ.