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) Xτ and Ø ∈ τ; (2) U, Vτ implies U ∩ V ∈ τ; and (3) If Uaτ for all a in some family (which could be infinite), then ∪a Uaτ.
* 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 τ1τ1 and join the topology generated by τ1τ2 as subbasis.
* Partial order: The topology τ on X is finer or stronger than the topology τ' if Uτ' implies Uτ; > 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 xX to consist of the open balls Br(x):= {yX | 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 (GP 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: IM 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||, with || A ||:= sup{|| Ax ||, || || ≤ 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 ∈ \(\cal H\), ∃ BS such that |\(\langle\)(AB) 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.

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