Topology, Topological Space| Topology, Topological Space |
In General > s.a. Combinatorial
Topology; Homeomorphism Problem.
$ Def: A topological
space is a pair (X, τ), with X a set and
τ a family of subsets of X,
called open sets, such that (1) X ∈ τ and Ø ∈ τ;
(2) U, V ∈ τ implies U ∩ V ∈ τ;
and (3) If Ua ∈ τ for
all a in some family (which could
be infinite), then ∪a
Ua ∈ τ.
* Remark: A good illustration of the
math program of isolating key abstract ideas.
* Areas of topology: See algebraic topology,
characteristic classes, knots.
Operations on One Topology > s.a. de Groot Dual.
* 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, τ),
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".
* Extension: A space Y
is called an extension of a space X if Y contains X
as a dense subspace; Two extensions of X are equivalent if there is
a homeomorphism between them which fixes X pointwise; For equivalence
classes of extensions [Y] and [Z], [Y] < [Z]
if there is a continuous function of Z into Y which fixes
X pointwise.
* Defined by additional structure:
Subset topology; Pullback topology; Quotient topology (e.g., T0 quotient;
notice: products and quotients do not necessarily "cancel out").
@ Extensions of topological spaces:
Mukherjee et al T&A(07) [ideal extension];
Koushesh T&A(11) [one-point extensions].
Space of Topologies > s.a. Adjunction Space;
Connected Sum; Inductive Limit;
join; projective limit;
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 τ1
∩ τ1 and join the topology generated by
τ1 ∪ τ2
as subbasis.
* Partial order: The topology τ
on X is finer or stronger than the topology τ'
if U ∈ τ'
implies U ∈ τ; > s.a
Wikipedia page.
* Set of all topological spaces:
(Actually, homeomorphism classes) Partially ordered by homeomorphic embedding.
* Operations on two topologies:
Union of topological spaces (trivial); Product topology; Induced topology on a subset.
@ 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].
And Other Structures > s.a. affine structure;
differentiable manifold; graph;
lattice; manifold; normed space;
symplectic geometry; Vector Space.
* On a (pseudo)metric space:
Given a pseudometric d on a space X, a topology is induced
by defining the basis of neighborhoods of each x ∈ X to
consist of the open balls Br(x):=
{y ∈ X | d(x,y)
< r}; The topology is T2 iff d is a metric.
* On a Riemannian manifold:
There are results on how to get a topology from a metric;
> see riemannian 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]);
> s.a. posets.
* On a set of paths: If (X, τ)
is a topological space, we can define the compact-open topology τ'
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]:= {γ | γ a
path in X such that γ(K) ⊂ U}
[@ Kelley 55;
Greenberg & Harper 81, p32].
* On causal curves between p and q:
A basis is given by T[U]:= {σ |
σ: I → M continuous,
causal, σ(0) = p, σ(1) = q,
σ(I) ⊂ U, open in M}; If M is
causal, it is equivalent to the Leray topology, otherwise it is not Hausdorff;
> s.a. lines.
* On \(\cal B\)(\(\cal H\)):
Norm or uniform topology (induced by d(A, B):=
|| A−B ||, with || A ||:=
sup{|| Ax ||, || x || ≤ 1});
Weak-operator topology (the closure of S ⊂ \(\cal B\)(\(\cal H\))
is S':= {A ∈ \(\cal B\)(\(\cal H\))
| for all ε, x1,
..., xn, y1,
..., yn ∈ \(\cal H\),
∃ B ∈ S such that |\(\langle\)(A−B)
xi, yi\(\rangle\)|
< ε for all i}).
* On sets with algebraic
operations: For example Lie groups, topological vector spaces.
@ General references: Harris CQG(00)gq/99 [on chronological sets];
Kallel mp/00 [curves in projective space];
Guerrero T&A(13) [domination by metric spaces].
@ And order:
Erné & Stege Ord(91);
Tholen T&A(09);
Campión et al T&A(09) [order-representability of topological spaces].
@ On spaces of functions / maps:
Georgiou & Iliadis Top(09) [admissible topologies];
Jordan T&A(10)
[coincidence of compact-open, Isbell, and natural topologies];
Dolecki & Mynard T&A(10) [Isbell topology];
> s.a. Compact-Open Topology;
metric tensors [topology on space of metrics].
Related Concepts
> see bundle; combinatorics;
Complex; dimension; Flag;
Homeomorphism; limit; Locale;
Presentation; Retraction.
> Results: see Annulus
Conjecture; Bolzano-Weierstrass Theorem;
Brouwer Theorem.
> Spaces, structures:
see Approach Space; Germ; sphere;
simplex; Supermanifold; types
of topologies; uniformity.
> Subsets: see Accumulation Point;
Base; Boundary;
cover; Dense Subset; loop;
Subbase.
> Generalizations: see
Choquet Space; Topos Theory;
operator algebras [non-commutative topology].
> Applications: see topology in physics.
General References
@ Texts, II: Mendelson 68;
Armstrong 83; Borges 00;
Mortad 16 [exercises and solutions].
@ 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;
Jameson 74;
Massey 75;
Császár 78;
Schurle 79;
Seifert & Threlfall 80;
Nagata 85;
Brown 88;
Engelking & Sieklucki 88;
Dolecki & Mynard 16 [based on convergence].
@ Texts, and geometry: Wall 72;
Sher & Daverman 02;
Reid & Szendrői 05 [II];
Sakai 13 [dimension theory, retracts, simplicial complexes, etc].
@ History, status: James ed-88;
Novikov mp/00-conf;
Pearl T&A(01),
T&A(04) [open problems].
@ Infinite-dimensional: Anderson ed-69;
van Mill 89.
@ Special emphasis: Steen & Seebach 78 [counterexamples];
Preuss 02 [convenient topology];
Naimpally & Peters 13 [applications, proximity spaces].
@ Invariants:
Gelfand & Tsygan CMP(92) [and localization];
Rudyak T&A(10) [topological complexity];
> s.a. torsion.
@ Approximate / fuzzy topology:
Schulman JMP(71);
Lowen 85.
@ Related topics: Thom in(70) [and linguistics];
Comfort NYAS(79);
Johnstone BAMS(83);
Taylor 86 [clones];
Francis 87 [pictures];
Frolík ed-88;
Preuss 88 [categorical];
Vickers 89;
Isham in(91) [introduction];
Trnková T&A(12) [clones];
> s.a. 3-manifolds [algorithmic topology].
> Online resources: see Internet Encyclopedia of Science
pages;
Low Dimensional Topology blog.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 4 jul 2018