Exercise · ZW-METRIC-SPACE-010 · ★★★ Advanced
If \((X,d)\) is a metric space and \(A\subseteq X\), what happens to \(d\) restricted to \(A\times A\)?
Choose one option.
Show answer
- B. \((A,d|_{A\times A})\) is always a metric space
Explanation
Symmetry, separation and the triangle inequality hold for all points of \(X\), in particular for those of \(A\).
Review the concept: Metric space
A metric space is a set \(X\) together with a distance \(d\) between its points: a nonnegative number for each pair, symmetric, zero only between a point and itself, and satisfying the triangle inequality.
\[\begin{aligned}d(x,y)&=0\iff x=y,\qquad d(x,y)=d(y,x),\\ d(x,z)&\le d(x,y)+d(y,z)\end{aligned}\]
More practice problems: metric space
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- ZW-METRIC-SPACE-005 In a metric space, if d(x,y)=0, what follows?
- ZW-METRIC-SPACE-006 Why is d(x,y)=|x^2-y^2| not a metric on ℝ?
- ZW-METRIC-SPACE-007 Let d(x,y)=(x-y)^2 on ℝ. For the points 0,1,2, what is d(0,2)-big(d(0,1)+d(1,2)big)?
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