Definition:Standard Discrete Metric
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Definition
The standard discrete metric on a set $S$ is the metric satisfying:
- $\map d {x, y} = \begin{cases} 0 & : x = y \\ 1 & : x \ne y \end{cases}$
This can be expressed using the Kronecker delta notation as:
- $\map d {x, y} = 1 - \delta_{x y}$
The resulting metric space $M = \struct {S, d}$ is the standard discrete metric space on $S$.
Special Cases
Real Number Plane
The discrete metric on $\R^2$ is defined as:
- $\displaystyle d_0 \left({x, y}\right) := \begin{cases} 0 & : x = y \\ 1 & : \exists i \in \left\{{1, 2}\right\}: x_i \ne y_i \end{cases}$
where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.
Also known as
This metric is also reported in some texts as the discrete metric.
Also see
- Results about the standard discrete metric can be found here.
Linguistic Note
Be careful with the word discrete.
A common homophone horror is to use the word discreet instead.
However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.
Sources
- 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 2.2$: Metric Spaces: Exercise $7$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{III}$: The Definition
- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 1 - 3: \ 6$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2.2$: Examples: Example $2.2.2$
- 1981: Karl R. Stromberg: An Introduction to Classical Real Analysis: $3.1$
- 1999: Theodore W. Gamelin and Robert Everist Greene: Introduction to Topology (2nd ed.) ... (previous) ... (next): $\S 1.1$: Open and Closed Sets: $(1.7)$