Metric spaces and neighborhoods

Try

Distance need not mean straight-line length in a drawing. On R², Euclidean distance uses a square root of summed squares, while taxicab distance adds absolute coordinate changes. Between (0,0) and (1,1), these distances are √2 and 2. Both support sensible neighborhoods and convergence, but their unit balls have different shapes. The common structure is expressed by the metric axioms.

Predict

Predict which familiar properties are indispensable. A metric d(x,y) must be nonnegative, vanish exactly when x=y, be symmetric and satisfy d(x,z)≤d(x,y)+d(y,z). Squared Euclidean distance fails the last condition: between zero and two on R its value is four, exceeding the sum one plus one through the intermediate point one. A visually plausible dissimilarity is not automatically a metric.

Experiment

Compare standard balls centered at zero in R².

Metric	Unit ball condition	Shape
Euclidean	x²+y²<1	Disk
Taxicab	|x|+|y|<1	Diamond
Maximum	max(|x|,|y|)<1	Axis-aligned square

Sketch all three with equal coordinate scales. The table supplies exact inequalities; the sketch illustrates them. A discrete metric on an arbitrary set gives another behavior: a ball of radius less than or equal to one contains only its center, regardless of any surrounding picture.

Understand

For the discrete metric, define distance zero for equal points and one otherwise. The triangle inequality holds because if x≠z, at least one of x≠y or y≠z must hold, so the two-step distance is at least one. The other axioms follow directly. This example shows that metric spaces can be sets without vector addition, slopes or line segments.

An open ball B(a,r) consists of points x with d(a,x)<r. Every such ball is open. For x inside it, choose ε=r−d(a,x)>0. If d(x,y)<ε, the triangle inequality gives d(a,y)<r. Thus a neighborhood around each member stays inside the ball. Arbitrary unions and finite intersections of open sets follow by the same neighborhood logic used on R, without any coordinate formulas.

Apply

A subset inherits the ambient metric. Its open sets are intersections with ambient open sets, but need not be open in the larger space. For example, [0,1) is open relative to [0,2] because it is the intersection with (−1,1); it is not open in R because every neighborhood of zero contains negative points.

Repair “different distance values always mean different open sets.” On R², ||v||∞≤||v||₂≤√2||v||∞. These inequalities let each metric ball contain a suitable ball of the other metric, so they generate the same topology. They do not assign the same distances or necessarily the same numerical Lipschitz constants.

Continue

Optional depth distinguishes metric structure from topology. Open sets determine continuity and convergence, but quantitative bounds may depend on the particular metric. Later examples will show that completeness can change when a metric is replaced by another giving the same open sets.

Save a full verification of one non-Euclidean metric and a failed triangle inequality for a nonexample. Then describe a subspace neighborhood without relying on a drawing. Next, sequence and continuity arguments will be rewritten using distances, revealing which Analysis I proofs genuinely used real-number order and which used only metric axioms.