The discrete topology on a set is the topology given by the power set of . That is, every subset of is open in the discrete topology. A space equipped with the discrete topology is called a discrete space.
The following conditions are equivalent:
is a discrete space.
Every subset of containing is a neighborhood of .
Suppose is a topological space and is a subset of . Then is a discrete subspace if and only if, for any , there is an open such that
, as a metric space with the standard distance metric , has the discrete topology.
The set of unit fractions , as a subspace of with the usual topology, is discrete. But is not, since any open set containing contains some unit fraction.
|Date of creation||2013-03-22 12:29:56|
|Last modified on||2013-03-22 12:29:56|
|Last modified by||mathcam (2727)|
|Synonym||discrete topological space|