Difference between revisions of "Arbitrarily close"
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Latest revision as of 17:56, 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 
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.)
or less from 
. (A set See also
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