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[parent] generalization of a pseudometric (Definition)

Let $ X$ be a set. Let $ d:X\times X \to \mathbb{R}$ be a function with the property that $ d(x,y)\ge 0$ for all $ x,y\in X$. Then $ d$ is a

  1. semi-pseudometric if $ d(x,y)=d(y,x)$ for all $ x,y\in X$,
  2. quasi-pseudometric if $ d(x,z)\le d(x,y)+d(y,z)$ for all $ x,y,z\in X$.

$ X$ equipped with a function $ d$ described above is called a semi-pseudometric space or a quasi-pseudometric space, depending on whether $ d$ is a semi-pseudometric or a quasi-pseudometric. A pseudometric is the same as a semi-pseudometric that is a quasi-pseudometric at the same time.

If $ d$ satisfies the property that $ d(x,y)=0$ implies $ x=y$, then $ d$ is called a semi-metric if $ d$ is a semi-pseudometric, or a quasi-metric if $ d$ is a quasi-pseudometric.



"generalization of a pseudometric" is owned by CWoo.
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See Also: semimetric, quasimetric, generalization of a uniformity

Other names:  semipseudometric, quasipseudometric, semipseudometric space, quasipseudometric space
Also defines:  semi-pseudometric space, quasi-pseudometric space, semi-pseudometric, quasi-pseudometric

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Cross-references: quasi-metric, implies, pseudometric, property, function
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This is version 3 of generalization of a pseudometric, born on 2007-02-20, modified 2007-02-20.
Object id is 8936, canonical name is GeneralizationOfAPseudometric.
Accessed 3069 times total.

Classification:
AMS MSC: 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)
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