Limits

In General > s.a. Filter; sequence.
$Topological def: The function f : XY tends to the limit y0 as xx0 if for any neighborhood V of y0 there is a neighborhood U of x0 such that if xU, then f(x) ∈ V.$ Metric space def: The function f : XY tends to the limit y0 as xx0 if for any positive ε there is a δ such that if d(x, x0) < δ then d(f(x), y0) < ε.
$Heine limit: The function f : XY has y0 as the Heine limit as xx0 if for any sequence {xn} converging to x0 in X, the sequence {f(xn)} converges to y0 in Y as n → ∞. > Online resources: see Wikipedia pages on limits of functions and limits of sequences. Infimum and Supremum Limits$ lim sup: Given a sequence {xi}i in $$\mathbb N$$, define am:= l.u.b. {xm, xm+1, ...}; Then lim supn → ∞ xn:= limm → ∞ am.

Direct or Inductive Limit > s.a. lie group; uniform space.
$Of topological spaces: Given the sequence X1X2 ⊂ ··· ⊂ Xi ⊂ ··· of topological spaces {(Xi, Ti)}, its inductive limit is the space X:= ∪i=1 Xi, with the topology XAT iff AXiTi, for all i. * Example: $$\mathbb C$$P is the inductive limit of $$\mathbb C$$P → $$\mathbb C$$P → ···.$ Of C*-algebras: Given an inductive family (I, {$$\cal A$$i}, {φij}), its inductive limit is the set of equivalence classes of "Cauchy sequences", $$\cal A$$:= { {ai}iI | ai ∈ $$\cal A$$i}, with || φij(ai) − aj ||m → 0 as i, j → ∞.
* Norm: || a ||:= limi → ∞ || ai ||i, for any representative family.
\$ Of posets: Given (I, {Pi}, {φij}), define P:= ∪iI Pi, with p < q iff there exists i in I such that p, qPi and p <i q.
@ General references: in Eilenberg & Steenrod 52; Fell & Doran 88; Murphy 90.
@ Poset completion: Meyer & Sorkin pr(89); in Bombelli & Meyer PLA(89); > s.a. the set of posets.
> Online resources: see Encyclopedia of Mathematics page; PlanetMath page; Wikipedia page.

Related Topics > see lorentzian geometry [limits of spacetimes]; projective limits.