Math 2.0

General topology Definition

Product metric

Builds on Metric spaceCartesian product

Definition

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces. The product metric on the Cartesian product X×YX \times Y is the function dd with d((x1,y1),(x2,y2))=max⁡(dX(x1,x2), dY(y1,y2))d\bigl((x_1, y_1), (x_2, y_2)\bigr) = \max\bigl(d_X(x_1, x_2), \, d_Y(y_1, y_2)\bigr) for all x1,x2∈Xx_1, x_2 \in X and y1,y2∈Yy_1, y_2 \in Y.

Remarks

The function dd is a metric on X×YX \times Y. Other common product metrics are dX+dYd_X + d_Y and dX2+dY2\sqrt{d_X^2 + d_Y^2}. Each of them is between dd and 2d2d, so all of them are equivalent and give the product topology. For X=Y=ℝX = Y = \mathbb{R} with the usual metric, dd is the sup metric on ℝ2\mathbb{R}^2. The same formula with a maximum over nn factors gives a product metric on a product of nn metric spaces.

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  1. Element definition
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  6. Predicate definition
  7. Ordered pair definition
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Function definition
  15. Metric space definition
  16. Product metric definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Product_metric, lead section
About this page
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Aliasesmax-product-metric
Kinddefinition
WritingW0 stub
MSC54E35
DefinesProduct metric
UsesMetric space, Cartesian product

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