Condensation point

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.
gollark: Try `www.theprism.net`.
gollark: I would really like clipboard capability for, say, pastebin put.
gollark: Which seems to me a bit of an oversight, but oh well.
gollark: Well, if you want to dynamically add new coroutines, parallel won't work.
gollark: You could give each module its own subcanvas so they don't mess with renders.

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


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.