discrete space
The discrete topology on a set is the topology given bythe 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 discrete topology is the http://planetmath.org/node/3290finest topology one can give to a set. Any set with the discrete topology is metrizable by defining for any with , and for any .
The following conditions are equivalent:
- 1.
is a discrete space.
- 2.
Every singleton in is an open set.
- 3.
Every subset of containing is a neighborhood
of .
Note that any bijection between discrete spaces is trivially a homeomorphism.
Discrete Subspaces
If is a subset of , and the subspace topology on is discrete, then is called a discrete subspace or discrete subset 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
Examples
- 1.
, as a metric space with the standard distance metric , has the discrete topology.
- 2.
, as a subspace
of or with the usual topology, is discrete. But , as a subspace of or with the trivial topology, is not discrete.
- 3.
, as a subspace of with the usual topology, is not discrete: any open set containing contains the intersection
of an open ball around with the rationals. By the Archimedean property, there’s a rational number
between and in . So can’t contain just : singletons can’t be open.
- 4.
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.
- 5.
The product
of two discrete spaces is discrete under the product topology. The product of an infinite
number of discrete spaces is discrete under the box topology, but if an infinite number of the spaces have more than one element
, it is not discrete under the product topology.
Title | discrete space |
Canonical name | DiscreteSpace |
Date of creation | 2013-03-22 12:29:56 |
Last modified on | 2013-03-22 12:29:56 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 17 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 54-00 |
Synonym | discrete topological space |
Related topic | Discrete2 |
Defines | discrete subspace |
Defines | discrete topology |
Defines | discrete space |
Defines | discrete subset |