Limit point

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A metric space $E$ is called complete if every fundamental sequence in $E$ converges to some element of the space. If each neighborhood of a point $f \subset E$ contains infinitely many points of a set $M$ in $E$, then $f$ is called a limit point of $M$. If a set contains all its limit points then it is said to be closed. The set consisting of $M$ and its limit points is called the closure of $M$ and is denoted by $\bar{M}$.[1]

If $A$ is a subset of the topological space $X$ and if $x$ is a point of $X$, then $x$ is a limit point of $A$ if every neighborhood of $x$ intersects $A$ in some point other than itself.[2]


  1. Achiezer, Naum I., and Izrail M. Glazman. Theory of Linear Operators in Hilbert Space. New York, NY: Dover, 1993.
  2. Munkres, James R. Topology, 2015.