Questions: Metric Topology

Question 1

The Euclidean metric d₂(x,y) = √(Σ(xᵢ−yᵢ)²) and the taxicab metric d₁(x,y) = Σ|xᵢ−yᵢ| assign different distances to the same pairs of points in ℝⁿ. What is the relationship between the topologies they induce?

• A. They induce different topologies, because the open balls have different shapes
• B. They induce the same topology, because every Euclidean open ball contains a taxicab open ball and vice versa
• C. Only the Euclidean metric induces a topology, because the taxicab metric is not smooth
• D. They induce the same topology only in ℝ¹, not in higher dimensions

Answer: (B)

The Euclidean and taxicab metrics are topologically equivalent: they generate exactly the same collection of open sets, even though their open balls look different (circles vs. diamonds). The key is that for any Euclidean ball B₂(x,r), you can find a taxicab ball B₁(x,r') contained in it, and vice versa — so every open set in one topology is also open in the other. This is the fundamental insight of metric topology: the topology depends on the *comparative* structure of distances, not their specific values.

Question 2

A student says: 'To understand whether a function f: X → Y between metric spaces is continuous, I need to know the exact distances assigned by the metrics, since continuity is an ε-δ condition.' What is the key limitation of this view?

• A. The student is correct — ε-δ continuity requires the specific metric values
• B. Continuity only depends on the topologies induced by the metrics, not the specific distance values; equivalent metrics give the same continuous functions
• C. Continuity between metric spaces cannot be defined using ε-δ at all
• D. The student is correct only for Hausdorff spaces

Answer: (B)

Continuity is a topological property: a function is continuous if and only if preimages of open sets are open. If two metrics induce the same topology, they determine exactly the same continuous functions — even though the ε and δ values you need to use in a proof may differ. This is why topology abstracts away from specific distances: the open set structure captures everything relevant to continuity, limits, and connectedness, while the metric values are extra data that may be redundant. Two topologically equivalent metrics are interchangeable for any question about continuous maps.

Question 3

Two metrics that induce the same topology on a space should assign the same distance to most pair of points.

• T. True
• F. False

Answer: False

False. Topological equivalence means the metrics generate the same open sets, not that they assign equal distances. The Euclidean metric and the taxicab metric on ℝ² assign different distances: d₂((0,0),(1,1)) = √2 while d₁((0,0),(1,1)) = 2. Yet both metrics induce the same topology on ℝ². The word 'equivalent' here means equivalent *for topological purposes* — identical open sets, same continuous functions, same limits — but not identical distance values.

Question 4

In the metric topology, sequences are sufficient to detect whether a point is a limit point or a set is closed.

• T. True
• F. False

Answer: True

True, and this follows from the metric topology being first-countable: each point has a countable neighborhood base (the balls of radius 1/n for n = 1, 2, 3, ...). In any first-countable space, sequential convergence characterizes closure and continuity — you don't need nets or filters. This is why real analysis can do everything with sequences: the Euclidean metric gives ℝ a first-countable topology. In non-metrizable topological spaces, sequences may be insufficient and nets become necessary.

Question 5

Why does topology focus on open sets rather than distances, and what does this shift in perspective gain?

Model answer: Topology focuses on open sets because open set structure captures everything relevant to continuity, connectedness, and convergence, while specific distance values are extra data that different metrics can vary without changing any of those properties. Two metrics that generate the same open sets define the same continuous functions, the same limits, and the same compact sets — making them interchangeable for any topological question. The gain is generality: once you know that continuity, homeomorphism, and compactness are purely topological concepts, you can study them in spaces that have no metric at all, extending the reach of analysis far beyond what metric-specific reasoning allows.

Metric topology is the bridge between analysis (which thinks in ε and δ) and general topology (which thinks in open sets). The realization that two different metrics can be topologically equivalent — differing in distances but agreeing on open sets — is the conceptual key that motivates abstract topology. Dropping the metric and keeping only the open set structure allows topology to apply to function spaces, quotient spaces, and infinite-dimensional spaces where no single metric is natural, while retaining all the tools needed to reason about continuity and convergence.