Topology,
Topological Space| Topology,
Topological Space |
In General > s.a. Combinatorial
Topology; Homeomorphism
Problem.
$ Def: A topological
space is a pair (X, 
),
with X a set and
a family of subsets
of X,
called open sets, such that (1) X 
and Ø ;
(2) U, V implies
U 
V
;
and (3) If Ua for
all a in some family (which could
be infinite), then 
a
Ua .
* Remark: A good illustration of the math program of isolating key abstract
ideas.
* Areas of topology: See algebraic topology, characteristic classes,
knots.
* Fineness: A partial
order; (X, )
is finer than (X', ')
if U '
implies
U .
Space of Topologies, New Topologies out of Old > s.a.
Tychonoff theorem.
* Set of topologies on a
set X:
Given a set, the set of topologies on it is partially ordered by fineness;
In fact, it is a lattice under inclusion, with meet 1 1
and join the topology generated by 1 2
as subbasis.
* Set of all topological
spaces:
(Actually, homeomorphism classes) Partially ordered by homeomorphic embedding.
* Operations on one topology:
Subset topology; Pullback topology; Quotient topology (e.g., T0 quotient;
notice that products and quotients do not necessarily "cancel out").
* Cone on a space: Given
a topological space X, the cone on X is
CX:= (X × I)/(X × {0}), with I:= [0,
1]; Properties:
For any X,
the cone CX is contractible.
* Suspension: Given a
compact (X, ),
the suspension S(X)
is homeomorphic to the topological space (X × [–1, 1])/
,
where is the equivalence
relation which identifies all points in X ×
{–1} and
all points in X × {1}; The suspension is like a "double
cone over X".
* Operations on two topologies: Union of topological spaces (trivial);
Product topology.
@ Set of topologies on X: Birkhoff FM(36);
Grib & Zapatrin IJTP(92),
IJTP(96)gq/95 [and
quantum logic]; Knight et al Ord(97);
Carlson T&A(07)
[Hausdorff topologies, lower/upper topologies].
@ Set of all topologies: Comfort & Gillam T&A(06)
[embeddability order].
@ Other constructions:
Mukherjee et al T&A(07)
[ideal extension]; > s.a. Adjunction
Space; Connected Sum; Inductive
Limit;
projective limit; join.
And Other Structures > s.a. affine
structure; differentiable
manifold; graph; lattice; manifold;
symplectic geometry; Vector
Space.
* On a metric/normed space: A topology is induced in any metric or
normed space.
* On a Riemannian manifold:
There are results on how to get a topology from a metric; > see geometry.
* On a Lorentzian manifold: Use the Alexandrov topology, or for compact
cases Johan's definition.
* On a poset: Interval
topology (closed intervals are a subbasis for closed sets); Order topology
(G 
P is
open if G eventually
contains any
net P which order-converges to an element of G [@ Birkhoff
67]).
* On a set of paths:
If (X, )
is a topological space, we can define the compact-open topology '
in the set of paths of X as follows; Consider K compact
I, and U open X;
Then a subbase is all sets [K, U]:=
{
| a
path in X such that (K) U}
[@ Kelley 55; Greenberg & Harper 81, p32].
* On causal curves between p and q:
A basis is given by T[U]:= {
|
: I → M continuous,
causal, (0) = p, (1)
= q,
(I) U,
open in M}; If M is
causal, it is equivalent to the Leray topology, otherwise it is not Hausdorff.
* On 
(
):
Norm or uniform topology (induced by d(A, B):= 
A–B ,
with
A :=
sup{ Ax , x 
1});
Weak-operator topology (the closure of S ()
is
S':= {A ()
| for all 
, x1,
..., xn, y1,
..., yn , 
B S such
that |
(A–B)
xi, yi
| < for
all i}).
* On sets with algebraic
operations: For example Lie groups, topological
vector spaces.
@ References: Erné & Stege Ord(91)
[and order]; Harris CQG(00)gq/99 [on
chronological
sets]; Kallel mp/00 [curves
in projective space].
Related Concepts > see Annulus
Conjecture; Bolzano-Weierstrass;
Brouwer Theorem; bundle; combinatorics;
Complex; dimension; Flag; Homeomorphism;
limit;
Locale; Presentation; Retraction.
> Spaces, structures:
see sphere; simplex; Supermanifold; types
of topologies; uniformity.
> Subsets: see Accumulation
Point;
Base; Boundary;
cover; Dense
Subset; loop;
Subbase.
> Generalizations: see
Choquet Space; Topos
Theory.
General References > s.a. topology
in physics.
@ Texts, II: Mendelson 68; Armstrong 83; Borges 00; Reid & Szendröi
05 [and geometry].
@ Texts: Lefschetz 30, 49; Kelley 55; Dugundji 60; Alexandroff 61; Bourbaki
61; Hocking & Young 61; Pervin 64; Singer & Thorpe 67; Engelking
68; Schubert 68; Porteous 69; in Maddox 70; Willard 70; Wall 72; Jameson 74;
Massey
75; Csaszar 78; Schurle 79; Seifert & Threlfall 80; Nagata 85;
Brown 88; Engelking & Sieklucki 88.
@ History, status: James ed-88; Novikov mp/00-in;
Pearl T&A(01), T&A(04)
[open problems].
@ Infinite-dimensional: Anderson 69; van Mill 89.
@ Special emphasis: Steen & Seebach 78 [counterexamples]; Preuss
02 [convenient topology].
@ Approximate/fuzzy topology: Schulman JMP(71);
Lowen 85.
@ Related topics: Thom in(70) [and linguistics]; Comfort NYAS(79);
Johnstone BAMS(83);
Taylor 86; Francis 87; Frolik ed-88; Preuss 88 [categorical]; Vickers 89; Isham
in(91) [introduction]; Gelfand & Tsygan CMP(92)
[invariants and localization].
> Online resources:
see Internet Encyclopedia of Science pages.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008