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| | ==Definition== |
| | In mathematics, a '''condensation point''' ''p'' of a subset ''S'' of a [[topological space]], is any point ''p'', such that every [[open neighborhood]] of ''p'' contains [[uncountably many]] points |
| | Thus, "condensation point" is synonymous with "<math>\aleph_1</math>-[[accumulation point]]". |
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| | ==Examples== |
| | *If ''S'' = (0,1) is the open unit interval, a subset of the [[real numbers]], then 0 is a condensation point of ''S''. |
| | *If ''S'' is an uncountable subset of a set ''X'' endowed with the [[indiscrete topology]], then any point ''p'' of ''X'' is a condensation point of ''X'' as the only open neighborhood of ''p'' is ''X'' itself. |
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| | ==References== |
| | * [[Walter Rudin]], ''Principles of Mathematical Analysis'', 3rd Edition, Chapter 2, exercise 27 |
| | * [[John C. Oxtoby]], ''Measure and Category'', 2nd Edition (1980), |
| | * [[Lynn Steen]] and [[J. Arthur Seebach, Jr.]], ''Counterexamples in Topology'', 2nd Edition, pg. 4 |
| | [[Category:Mathematical objects]] |
| | [[Category:Topology]] |
Definition
In mathematics, a condensation point p of a subset S of a topological space, is any point p, such that every open neighborhood of p contains uncountably many points
Thus, "condensation point" is synonymous with "-accumulation point".
Examples
- If S = (0,1) is the open unit interval, a subset of the real numbers, then 0 is a condensation point of S.
- If S is an uncountable subset of a set X endowed with the indiscrete topology, then any point p of X is a condensation point of X as the only open neighborhood of p is X itself.
References
- Walter Rudin, Principles of Mathematical Analysis, 3rd Edition, Chapter 2, exercise 27
- John C. Oxtoby, Measure and Category, 2nd Edition (1980),
- Lynn Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, 2nd Edition, pg. 4