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$\varepsilon$-net (Definition)

Definition Suppose $ X$ is a metric space with a metric $ d$, and suppose $ S$ is a subset of $ X$. Let $ \varepsilon$ be a positive real number. A subset $ N\subset S$ is an $ \varepsilon$-net for $ S$ if, for all $ x\in S$, there is an $ y\in N$, such that $ d(x,y)<\varepsilon$.

For any $ \varepsilon>0$ and $ S\subset X$, the set $ S$ is trivially an $ \varepsilon$-net for itself.

Theorem Suppose $ X$ is a metric space with a metric $ d$, and suppose $ S$ is a subset of $ X$. Let $ \varepsilon$ be a positive real number. Then $ N$ is an $ \varepsilon$-net for $ S$, if and only if

$\displaystyle \{ B_\varepsilon(y) \mid y\in N \}$
is a cover for $ S$. (Here $ B_\varepsilon(x)$ is the open ball with center $ x$ and radius $ \varepsilon$.)

Proof. Suppose $ N$ is an $ \varepsilon$-net for $ S$. If $ x\in S$, there is an $ y\in N$ such that $ x\in B_\varepsilon(y)$. Thus, $ x$ is covered by some set in $ \{ B_\varepsilon(x) \mid x\in N \}$. Conversely, suppose $ \{ B_\varepsilon(y) \mid y\in N \}$ is a cover for $ S$, and suppose $ x\in S$. By assumption, there is an $ y\in N$, such that $ x\in B_\varepsilon(y)$. Hence $ d(x,y)<\varepsilon$ with $ y\in N$. $ \Box$

Example In $ X=\mathbb{R}^2$ with the usual Cartesian metric, the set

$\displaystyle N = \{(a,b) \mid a,b\in \mathbb{Z} \}$
is an $ \varepsilon$-net for $ X$ assuming that $ \varepsilon> \sqrt{2}/2$. $ \Box$

The above definition and example can be found in [1], page 64-65.

Bibliography

1
G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.



"$\varepsilon$-net" is owned by Koro. [ owner history (1) ]
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Cross-references: proof, radius, center, open ball, cover, real number, positive, subset, metric, metric space
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This is version 1 of $\varepsilon$-net, born on 2003-05-12.
Object id is 4280, canonical name is VarepsilonNet.
Accessed 3222 times total.

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