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example of pseudometric space
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(Example)
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Let $X=\mathbb{R}^2$ and consider the function $d:X\times X$ to the non-negative real numbers given by
Then $d(x,x)=|x_1-x_1|=0$ , $d(x,y)=|x_1-y_1|=|y_1-x_1|=d(y,z)$ and the triangle inequality follows from the triangle inequality on $\mathbb{R}^1$ , so $(X,d)$ satisfies the defining conditions of a pseudometric space.
Note, however, that this is not an example of a metric space, since we can have two distinct points that are distance 0 from each other, e.g.
Other examples:
- Let $X$ be a set, $x_0\in X$ , and let $F(X)$ be functions $X\to R$ . Then $d(f,g)=|f(x_0)-g(x_0)|$ is a pseudometric on $F(X)$ [1].
- If $X$ is a vector space and $p$ is a seminorm over $X$ , then $d(x,y)=p(x-y)$ is a pseudometric on $X$ .
- The trivial pseudometric $d(x,y)=0$ for all $x,y\in X$ is a pseudometric.
- 1
- S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
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"example of pseudometric space" is owned by mathcam. [ full author list (2) ]
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Cross-references: seminorm, vector space, pseudometric, distance, points, metric space, pseudometric space, triangle inequality, real numbers, function
This is version 3 of example of pseudometric space, born on 2004-10-02, modified 2007-05-26.
Object id is 6275, canonical name is ExampleOfPseudometricSpace.
Accessed 2819 times total.
Classification:
| AMS MSC: | 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability) |
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Pending Errata and Addenda
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