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: Now, of course, the servers it finds may not or no longer contain the site code, in which case it can try data recovery operations or piece it together from logs.
gollark: In parallel, it can attempt to access governmental IP traffic logs and find historical copies of the site to use.
gollark: If no exploitable vulnerabilities are found the next step is to launch a physical layer assault to access the servers in question.
gollark: It's permitted by GTech™ GRFC™ 11084G.
gollark: Now, if it can't find some humans to fix the site, the next step is to scan for exploitable vulnerabilities in related components to get access to the code and work out how to display the site.

References
  1. https://www.encyclopediaofmath.org/index.php/Condensation_point_of_a_set
  • 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


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