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A point x is a limit point of A if every neighborhood of x contains a point of A different from x. The derived set A' is the set of all limit points of A. Then cl(A) = A ∪ A'. A set is closed iff A = cl(A) iff A' ⊆ A. Limit points generalize the notion of convergence to arbitrary topologies.
A point x is a limit point (or accumulation point) of a set A in a topological space (X, τ) if every neighborhood of x contains at least one point of A that is different from x. Formally, for every open set U containing x, the intersection U ∩ (A \ {x}) is nonempty. The requirement that the point be different from x is essential: without it, every element of A would trivially be a limit point of A (since x ∈ U ∩ A whenever x ∈ A), and the concept would collapse. The "different from x" clause is what distinguishes genuine accumulation — being approached by other points of A — from mere membership.
The canonical example is the set A = {1/n : n = 1, 2, 3, ...} in ℝ. The point 0 is a limit point of A: every open interval around 0 contains 1/n for sufficiently large n. But 0 is not in A. Meanwhile, 1/2 ∈ A is not a limit point of A, because the interval (1/3, 2/3) contains 1/2 but no other element of A. A point of A that is not a limit point of A is called an isolated point — it sits alone with a neighborhood containing no other element of the set. In this example, every element of A is isolated, and the only limit point is 0.
The set of all limit points of A is called the derived set, denoted A'. The closure of A is then cl(A) = A ∪ A' — the original set together with all its limit points. This gives a characterization of closed sets: A is closed if and only if A' ⊆ A, meaning A contains all its limit points. If any limit point of A is missing from A, then A fails to be closed. This connects the abstract definition of closed (complement is open) to the intuitive notion of a set that "includes its boundary." The closure cl(A) is always closed, and it is the smallest closed set containing A.
In metric spaces and first-countable spaces, limit points can be detected by sequences: x is a limit point of A if and only if there exists a sequence of distinct points in A converging to x. But in general topological spaces, this sequential characterization can fail. A point can be a limit point of A without any sequence from A converging to it — the topology may have "too many" neighborhoods for sequences (indexed by ℕ) to probe. In such spaces, nets (generalized sequences indexed by directed sets) or filters are needed to capture all limit points. The definition via neighborhoods — every neighborhood of x meets A \ {x} — works universally, which is why it is the foundational one.