Discrete space: Difference between revisions
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In [[topology]], a '''discrete space''' is a topological space with the discrete topology, in which every [[subset]] is open. | {{subpages}} | ||
In [[topology]], a '''discrete space''' is a [[topological space]] with the '''discrete topology''', in which every [[subset]] is open. | |||
==Properties== | ==Properties== | ||
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* 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 }} | ||
Latest 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.