Difference between revisions of "Arbitrarily close"
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Latest revision as of 16:56, 28 March 2009
Informally, a set contains points arbitrarily close to some point
if, for any positive distance
, there are members of
which are less than a distance
from
.
More formally, in a metric space (such as the Euclidean space
or just the real numbers) with distance function
, a set
is said to contain members arbitrarily close to some point
if for all
there exists some
such that
.
Examples
- In the particular case of the real numbers with the usual distance
, the set
contains points arbitrarily close to 0.
- The set of real numbers contains points arbitarily close to any given real number. This is also true of the rational numbers, but it is not true of the integers. We can see this last fact is true because (for example) there is no integer at distance
or less from
. (A set
which contains points arbitrarily close to every point is said to be dense.)
See also
This article is a stub. Help us out by expanding it.