mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a condensation point ''p'' of a

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...

''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 points ...

is any point ''p'' such that every

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

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 contai ...

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 real ...

, then 0 is a condensation point of ''S''. *If ''S'' is an uncountable subset of a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''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 19 ...

, ''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. J. Arthur Seebach Jr (May 17, 1938 – December 3, 1996) was an American mathematician. Seebach studied Greek language as an undergraduate, making it a second major with mathematics. Seebach studied with A. I. Weinzweig at Northwestern Univ ...

, ''Counterexamples in Topology'', 2nd Edition, pg. 4 Mathematical objects Topology {{topology-stub