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 *U*_{a} ∈ *τ* for
all *a* in some family (which could
be infinite), then ∪_{a}
*U*_{a} ∈ *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
C*X*:= (*X* × I)/(*X* × {0}), with I:= [0,
1]; __Properties__:
For any *X*, the cone C*X* 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., T_{0} 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

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**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 B_{r}(*x*):=
{*y* ∈ *X* | *d*(*x*,*y*)
< *r*}; The topology is T_{2} 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

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**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.

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