A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
A point x is a limit point of a set A if every neighborhood of x contains points of A other than x itself. Convergence generalizes the real-line notion: a sequence converges to x if for every neighborhood of x, all but finitely many terms lie in that neighborhood. In general topological spaces, limits may not be unique, which is addressed by separation axioms.
You know what a neighborhood is in a topological space: an open set containing a point. Now we use neighborhoods to generalize two fundamental concepts from real analysis — limit points and sequence convergence — to arbitrary topological spaces. The key move is replacing "within ε of x" with "in every neighborhood of x," which works in any topological space, not just metric ones.
A point x is a limit point (also called an accumulation point) of a set A if every neighborhood of x contains at least one point of A *other than x itself*. The "other than x" clause is essential: it excludes isolated points that happen to be in A. Consider A = {0} ∪ (1, 2) in ℝ. The point 0 is isolated in A — the neighborhood (−0.5, 0.5) contains no other point of A — so 0 is not a limit point of A. But every point in the closed interval [1, 2] is a limit point of A: any neighborhood of such a point intersects (1, 2) in a nonempty open interval containing infinitely many points. The closure of A is then A together with all its limit points; a set is closed if and only if it contains all of its limit points. This gives a purely neighborhood-based way to compute closures without invoking distances.
Convergence in a topological space takes the same neighborhood approach: a sequence (xₙ) converges to x if for every open neighborhood U of x, there exists N such that xₙ ∈ U for all n > N. In a metric space, this reduces exactly to the standard ε definition (let U = B(x, ε)). But in a general topological space, a striking pathology can occur: limits need not be unique. In the indiscrete topology (where the only open sets are ∅ and X itself), every sequence converges to every point simultaneously, because the only neighborhood of any point is all of X, which trivially contains every term. This seems absurd, and it is — which is why Hausdorff spaces (T₂ spaces) are so important: a space is Hausdorff if any two distinct points have disjoint open neighborhoods, and in a Hausdorff space limits are always unique. The progression from general spaces to Hausdorff spaces mirrors the progression from pathological to well-behaved, and most spaces arising in practice are Hausdorff.