Indiscrete space: Difference between revisions
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In [[topology]], an '''indiscrete space''' is a [[topological space]] with the '''indiscrete topology''', in which the only open [[subset]]s are the empty subset and the space itself. | |||
==Properties== | |||
* An indiscrete space is [[metric space|metrizable]] if and only if it has at most one point | |||
* An indiscrete space is [[compact space|compact]]. | |||
* An indiscrete space is [[connected space|connected]]. | |||
* Every map from a topological space to an indiscrete space is [[continuous map|continuous]]. | |||
==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 }} | |||
Revision as of 16:04, 4 January 2013
In topology, an indiscrete space is a topological space with the indiscrete topology, in which the only open subsets are the empty subset and the space itself.
Properties
- An indiscrete space is metrizable if and only if it has at most one point
- An indiscrete space is compact.
- An indiscrete space is connected.
- Every map from a topological space to an indiscrete space is continuous.
References
- Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag. ISBN 0-387-90312-7.