View source for Condensation point of a set

''in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244401.png" />''

A point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244402.png" /> such that every neighbourhood of it contains uncountable many points of the set. The set of condensation points of a set is always closed; if it is non-empty, it is perfect and has the cardinality of the continuum. The concept of a condensation point can be generalized to arbitrary topological spaces.



====Comments====
The generalization to arbitrary spaces is direct: A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244403.png" /> a condensation point (of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244404.png" />) in a topological space if (the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244405.png" /> with) every neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024440/c0244406.png" /> is an uncountable set. (See also [[#References|[a1]]].)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>