Discrete space: Difference between revisions
Jump to navigation
Jump to search
imported>Howard C. Berkowitz No edit summary |
imported>Richard Pinch (Every map from a discrete space to a topological space is continuous) |
||
| Line 7: | Line 7: | ||
* A discrete space is [[compact space|compact]] if and only if it is [[finite set|finite]]. | * A discrete space is [[compact space|compact]] if and only if it is [[finite set|finite]]. | ||
* A discrete space is [[connected space|connected]] if and only if it has at most one point. | * A discrete space is [[connected space|connected]] if and only if it has at most one point. | ||
* Every map from a discrete space to a topological space is [[continuous map|continuous]]. | |||
==References== | ==References== | ||
* {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 | pages=41-42 }} | * {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 | pages=41-42 }} | ||
Revision as of 15:57, 4 January 2013
In topology, a discrete space is a topological space with the discrete topology, in which every subset is open.
Properties
- A discrete space is metrizable, with the topology induced by the discrete metric.
- A discrete space is a uniform space with the discrete uniformity.
- A discrete space is compact if and only if it is finite.
- A discrete space is connected if and only if it has at most one point.
- Every map from a discrete space to a topological space is continuous.
References
- Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 41-42. ISBN 0-387-90312-7.