In mathematics, a condensation point ''p'' of a

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

''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 In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...

Examples

*If ''S'' = (0,1) is the open unit interval, a subset of the

real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

, 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 1 ...

, ''Principles of Mathematical Analysis'', 3rd Edition, Chapter 2, exercise 27 *

John C. Oxtoby John C. Oxtoby (1910–1991) was an American mathematician. In 1936, he graduated with a Master of Science in Mathematics from Harvard University. He was professor of mathematics at Bryn Mawr College in Pennsylvania Pennsylvania (; ( Pennsy ...

, ''Measure and Category'', 2nd Edition (1980), * Lynn Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology'', 2nd Edition, pg. 4 Mathematical objects Topology {{topology-stub