Questions: The Topology Induced by a Metric

Question 1

On ℝ², the Euclidean metric generates circular open balls while the taxicab metric generates diamond-shaped open balls. What can you conclude about the topologies these two metrics generate?

• A. They generate different topologies because their open balls have different shapes
• B. They generate the same topology — the standard topology on ℝ² — because every open ball in one metric contains an open ball in the other around each point
• C. The taxicab metric generates a strictly coarser topology than the Euclidean metric
• D. Only the Euclidean metric generates a valid topology; the taxicab metric's open regions don't satisfy the topological axioms

Answer: (B)

Two metrics are topologically equivalent when every open ball in one metric contains an open ball in the other around every point. For the Euclidean and taxicab metrics on ℝ², this holds: you can always fit a taxicab diamond inside a Euclidean circle and vice versa (by shrinking the radius). Both generate the standard topology on ℝ², even though the geometric shape of their 'balls' differs. Topology retains only the qualitative structure of neighborhoods, not their exact shape.

Question 2

A student claims: 'The open balls B(x, ε) in a metric space form a topology on X.' What is wrong with this claim?

• A. Open balls don't contain their center point, so they cannot be open sets
• B. The claim is correct — the collection of all open balls is exactly the metric topology
• C. Open balls form a *basis* for a topology, but not a topology itself, because finite intersections of open balls are not always open balls
• D. Open balls only generate a topology if X is a subset of ℝⁿ

Answer: (C)

The collection of open balls is a *basis* for the metric topology, not the topology itself. A topology must be closed under finite intersections, but the intersection of two open balls of different sizes and centers is typically a lens-shaped region — not itself an open ball. The topology consists of all *unions* of open balls. This is why 'basis' is the right vocabulary: every open set in the topology is a union of basis elements, but the basis elements themselves don't form a topology.

Question 3

Most topology on a set X is induced by some metric on X.

• T. True
• F. False

Answer: False

Not every topology is metrizable. Metrization theorems (such as Urysohn's metrization theorem) characterize which topological spaces can be given a metric that generates their topology — conditions like second-countability and regularity are required. The indiscrete topology on a set with more than one point, for example, is not metrizable: any metric would generate a finer topology (open balls separate points), but the indiscrete topology has only ∅ and X as open sets.

Question 4

Two metrics that induce the same topology are called topologically equivalent, even if the exact distances they assign to pairs of points differ significantly.

• T. True
• F. False

Answer: True

Topological equivalence means the metrics generate the same open sets — the same neighborhoods, the same convergent sequences, the same continuous functions. The exact distances are irrelevant to the topology. On ℝⁿ, the Euclidean, taxicab, and maximum (L∞) metrics are all topologically equivalent: they differ on distances but agree on which sets are open. This is the sense in which topology 'abstracts away' geometry while retaining qualitative structure.

Question 5

Explain why the fact that different metrics can generate the same topology shows that topology 'abstracts away' something from metric geometry. What information is retained, and what is discarded?

Model answer: Metric geometry encodes exact distances: d(x,y) tells you precisely how far apart x and y are. Topology retains only the qualitative notion of 'nearness' — which sets count as neighborhoods, which sequences converge, which functions are continuous — without caring about the specific distance values. When two metrics generate the same topology, all topological properties (continuity, compactness, connectedness) are identical, even though distances differ. What is discarded: exact lengths, angles, and the shape of 'balls.' What is retained: the structure of open sets and the qualitative notion of points being 'close.'

This abstraction is what makes topology so powerful: results proved for metric spaces using only topological properties (open/closed sets, neighborhoods) apply to all equivalent metrics at once. Conversely, properties that depend on exact distances (like 'the distance between x and y is 5') are purely metric, not topological, and don't transfer. Metrization theory is the study of exactly where the line falls — which topological spaces retain enough structure to be given a compatible metric.