Condensation point
Appearance
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (April 2014) |
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 of S. Thus, "condensation point" is synonymous with "-accumulation point".[1]
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