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 page.

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.
> Online resources: see Wikipedia page.

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.


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