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In mathematics, a condensation point ''p'' of a subset ''S'' of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is any point ''p'' such that every neighborhood of ''p'' contains uncountably many points of ''S''. Thus "condensation point" is synonymous with "\aleph_1- 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 In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
, then any point ''p'' of ''X'' is a condensation point of ''X'' as the only neighborhood of ''p'' is ''X'' itself.


References

*
Walter Rudin Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Walter Jacobs (1930–1968) * Gunther (wrestler), Austrian professional wrestler and trainer Walter Hahn (born ...
, ''Principles of Mathematical Analysis'', 3rd Edition, Chapter 2, exercise 27 * John C. Oxtoby, ''Measure and Category'', 2nd Edition (1980), *
Lynn Steen Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a Professor of Mathematics at St. Olaf College, Northfield, Minnesota in the U.S. He wrote numerous books and articles on the teaching of mathematics. H ...
and J. Arthur Seebach, Jr., ''Counterexamples in Topology'', 2nd Edition, pg. 4 Mathematical objects Topology {{topology-stub