Limits |

**In General**
> s.a. Filter; sequence.

$ __Topological def__: The
function *f *: *X* → *Y* tends to the limit
*y*_{0} as *x* →
*x*_{0} if for any neighborhood
*V* of *y*_{0} there is
a neighborhood *U* of *x*_{0}
such that if *x* ∈ *U*, then *f*(*x*)
∈ *V*.

$ __Metric space def__: The
function *f* : *X* → *Y* tends to the limit
*y*_{0} as *x* →
*x*_{0} if for any positive
*ε* there is a *δ* such that if
*d*(*x*, *x*_{0})
< *δ* then *d*(*f*(*x*),
*y*_{0}) < *ε*.

$ __Heine limit__: The function
*f* : *X* → *Y* has *y*_{0}
as the Heine limit as *x* → *x*_{0}
if for any sequence {*x*_{n}} converging
to *x*_{0} in *X*, the sequence
{*f*(*x*_{n})} converges to
*y*_{0} in *Y* as *n* → ∞.

> __Online resources__:
see Wikipedia page.

**Infimum and Supremum Limits**

$ __lim sup__: Given a sequence
{*x*_{i}}_{i
in \(\mathbb N\)}, define
*a*_{m}:=
l.u.b. {*x*_{m},
*x*_{m+1}, ...};
Then lim sup_{n → ∞}
*x*_{n}:=
lim_{m → ∞}
*a*_{m}.

> __Online resources__:
see Wikipedia page.

**Direct or Inductive Limit**
> s.a. lie group; uniform space.

$ __Of topological spaces__:
Given the sequence *X*_{1} ⊂
*X*_{2} ⊂ ···
⊂ *X*_{i} ⊂ ···
of topological spaces {(*X*_{i},
*T*_{i})}, its inductive limit is the space
*X*:= ∪_{i=1}^{∞}
*X*_{i}, with the topology
*X* ⊃ *A* ∈ *T* iff *A* ∩
*X*_{i} ∈
*T*_{i}, 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}},
{*φ*_{ij}}), its inductive
limit is the set of equivalence classes of "Cauchy sequences",
\(\cal A\)_{∞}:=
{ {*a*_{i}}_{i
∈ I} | *a*_{i}
∈ \(\cal A\)_{i}}, with
|| *φ*_{ij}(*a*_{i})
− *a*_{j} ||_{m}
→ 0 as *i*, *j* → ∞.

* __Norm__:
|| *a* ||_{∞}:=
lim_{i → ∞}
|| *a*_{i}
||_{i}, for any representative family.

$ __Of posets__: Given (*I*,
{*P*_{i}},
{*φ*_{ij}}),
define *P*_{∞}:=
∪_{i ∈ I}
*P*_{i}, with
*p* <_{∞} *q*
iff there exists *i* in *I* such that *p*,
*q* ∈ *P*_{i}
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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