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nLabsubspace topology
Contents
- Definition
- Examples
- Properties
- Universal property
- General
- Variations
- Related concepts
- References
Definition
Definition
(subspace topology)
Nội dung chính
- nLabsubspace topology
- Universal property
- Related concepts
- The open subsets of a closed interval [0,1][0,1] subset mathbbR regarded as a topological subspace of the real line equipped with its Euclidean metric topology is generated from the sub-base β=[0,a),(a,1]a[0,1]beta = [0, a) ,(a,1]_a in [0,1].
- A subset of a metric space equipped with its induced metric is a subspace with respect to the corresponding metric topologies.
For ZZ any topological space, a function Z⟶fUZ oversetflongrightarrow U (of underlying sets) is continuous precisely if the composition ifi circ f is continuous as a function to XX:
Z⟶fUifiXarray Z &oversetflongrightarrow& U \ &_mathllapi circ fsearrow& Bigdownarrow^mathrlapi \ && X
embedding
embedding of topological spaces
embedding of smooth manifolds
Nicolas Bourbaki, Elements of Mathematics General topology, 1971, 1990
James Munkres, Topology, Prentice Hall (1975, 2000)
John Terilla, Notes on categories, the subspace topology and the product topology 2014 (pdf)
Mike Shulman, Brouwers fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2022): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)
Let (X,τX)(X, tau_X) be a topological space, and let YXY subset X be a subset of its underlying set. Then the corresponding topological subspace has YY as its underlying set, and its open subsets are those subsets of YY which arise as restrictions of open subsets of XX (i.e. intersections of open subsets of XX with YY):
(UYYopen)(UXτX(UY=UXY)).left( U_Y subset Y,,textopen right) ,Leftrightarrow, left( undersetU_X in tau_Xexists left( U_Y = U_X cap Y right) right) ,.
In other words, τYtau_Y is the smallest topology on YY such that the inclusion YXY hookrightarrow X is continuous (the initial topology on that map).
The picture on the right shows two open subsets inside the square, regarded as a topological subspace of the plane 2mathbbR^2:
graphics grabbed from Munkres 75
The pair (Y,τY)(Y,tau_Y) is then said to be a topological subspace of (X,τX)(X,tau_X). The induced topology is for that reason sometimes called the subspace topology on YY.
A continuous function that factors as a homeomorphism onto its image equipped with the subspace topology is called an embedding of topological spaces. Such a map is referred to as a subspace inclusion.
A property of topological spaces is said to be hereditary if its satisfaction for a topological space XX implies its satisfaction for all topological subspaces of XX.
Examples
The image on the right shows open subsets in the closed square [0,1]2[0,1]^2, regarded as a topological subspace of the Euclidean plane
Properties
Universal property
Proposition
(universal property of subspace topology)
Let U⟶iXU oversetilongrightarrow X be an injective continuous function between topological spaces. Then this is a subspace inclusion (Def. ) precisely if it satisfies the following universal property:
The elementary proof is spelled out, for instance, in Terilla 14, theorem 1. Of course this is just another way to speak of the initial topology.
The universal characterization of Prop. lends itself to formalization via axioms for cohesion:
Definition
(sharp modality on topological spaces)
Let
Set⟶coDisc⟵ΓTopSet underoverset undersetcoDisclongrightarrow oversetGammalongleftarrow bot Top
be the pair of adjoint functors given by sending a topological space XX to its underlying set Γ(X)Gamma(X), and by equipping a set SS with the codiscrete topology making it a codiscrete space coDisc(X)coDisc(X).
Write
coDiscΓ:Top⟶Topsharp ;coloneqq; coDisc circ Gamma ;colon; Top longrightarrow Top
for the induced modal operator on Top (sharp modality). We write
id⟶ηid overseteta^sharplongrightarrow sharp
for the unit morphism of this adjunction.
Notice that this means that for any topological space ZZ, every function of underlying sets
Z⟶XZ longrightarrow sharp X
is continuous functions, hence that continuous functions into Xsharp X are in natural bijection to underlying functions of sets. This is the statement of the adjunction hom-isomorphism:
HomTop(Z,X)HomSet(Γ(Z),Γ(X)).Hom_Top( Z, sharp X ) ;simeq; Hom_Set(Gamma(Z), Gamma(X)) ,.Proposition
Let U⟶iXU oversetilongrightarrow X be an injective continuous function between topological spaces. Then this is a subspace inclusion (Def. ) precisely if its naturality square of the sharp-unit (Def. )
U⟶ηUUiiX⟶ηXXarray U &overset eta^sharp_U longrightarrow& sharp U \ ^mathllapiBigdownarrow && Bigdownarrow^mathrlapsharp i \ X &underseteta^sharp_Xlongrightarrow& sharp X
is a pullback square.
Proof
By the universal property of a pullback/fiber product and the nature of sharp, we have UX×XUU simeq X times_sharp X sharp U precisely if continuous functions out of some topological space ZZ into UU are in natural bijection with continuous functions ZXZ to X whose underlying function ZXXZ to X to sharp X factors through the underlying function of ii. This implies the statement by Prop. .
Remark
(formulation in cohesive homotopy type theory)
The pullback square of the sharp-unit in Prop. should correspond (after generalizing from topological spaces to suitable topological -groupoids) to the categorical semantics of what in cohesive homotopy type theory is the statement that the characteristic function
χU:XPropchi_U ;colon; X to Prop
to the universe of propositions factors through the universe of sharp-modal types. In this form topological subspace inclusions are characterized in Shulman 15, Remark 3.14.
General
A subspace i:YXi: Y hookrightarrow X is closed if YY is closed as a subset of XX (or if ii is a closed map), and is open if YY is open as a subset of XX (or if ii is an open map).
Proposition
Topological subspace inclusions (topological embedding) are precisely the regular monomorphisms in the category Top of all topological spaces.
For example, the equalizer of two maps f,g:XYf, g colon X stackreltoto Y in Top is computed as the equalizer the underlying-set level, equipped with the subspace topology.
Lemma
The pushout in Top of any (closed/open) subspace i:ABi colon A hookrightarrow B along any continuous function f:ACf colon A to C,
AiBfpogCjD,array A & stackrelihookrightarrow & B \ mathllapf downarrow & po & downarrow mathrlapg \ C & undersetjhookrightarrow & D,
is a (closed/open) subspace j:CDj: C hookrightarrow D.
Proof
Since U=hom(1,):TopSetU = hom(1, -): Top to Set is faithful, we have that monos are reflected by UU; also monos and pushouts are preserved by UU since UU has both a left adjoint and a right adjoint. In SetSet, the pushout of a mono along any map is a mono, so we conclude jj is monic in TopTop. Furthermore, such a pushout diagram in SetSet is also a pullback, so that we have the Beck-Chevalley equality if*=g*j:P(C)P(B)exists_i circ f^ast = g^ast exists_j colon P(C) to P(B) (where i:P(A)P(B)exists_i colon P(A) to P(B) is the direct image map between power sets, and f*:P(C)P(A)f^ast: P(C) to P(A) is the inverse image map).
To prove that jj is a subspace, let UCU subseteq C be any open set. Then there exists open VBV subseteq B such that i*(V)=f*(U)i^ast(V) = f^ast(U) because ii is a subspace inclusion. If χU:C2chi_U colon C to mathbf2 and χV:B2chi_V colon B to mathbf2 are the maps to Sierpinski space that classify these open sets, then by the universal property of the pushout, there exists a unique continuous map χW:D2chi_W colon D to mathbf2 which extends the pair of maps χU,χVchi_U, chi_V. It follows that j1(W)=Uj^-1(W) = U, so that jj is a subspace inclusion.
If moreover ii is an open inclusion, then for any open UCU subseteq C we have that j*(j(U))=Uj^ast(exists_j(U)) = U (since jj is monic) and (by Beck-Chevalley) g*(j(U))=i(f*(U))g^ast(exists_j(U)) = exists_i(f^ast(U)) is open in BB. By the definition of the topology on DD, it follows that j(U)exists_j(U) is open, so that jj is an open inclusion. The same proof, replacing the word open with the word closed throughout, shows that the pushout of a closed inclusion ii is a closed inclusion jj.
A similar (but even simpler) line of argument establishes the following result.
Lemma
Let κkappa be an ordinal, viewed as a preorder category, and let F:κTopF: kappa to Top be a functor that preserves directed colimits. Then if F(ij)F(i leq j) is a (closed/open) subspace inclusion for each morphism iji leq j of κkappa, then the canonical map F(0)colimiκF(i)F(0) to colim_i in kappa F(i) is also a (closed/open) inclusion.
Variations
There is also a notion of a Grothendieck topology induced along a functor from a Grothendieck topology on another category (actually the input can be a somewhat more general coverage, then the topology induced along the identity functor will serve as a sort of a completion). (this will be explained later).
A topology may be induced by more than a function other than a subset inclusion, or indeed by a family of functions out of YY (not necessarily all with the same target). However, the term induced topology is often (usually?) restricted to subspaces; the general concept is called a weak topology. (This construction can be done in any topological concrete category; in this generality it is often called an initial structure for a source.) The dual construction (involving functions to YY) is a strong topology (or final structure for a sink); an example is the quotient topology on a quotient space.
Related concepts
examples of universal constructions of topological spaces:
AAAAphantomAAAAlimitsAAAAphantomAAAAcolimits, point space,, empty space ,, product topological space ,, disjoint union topological space ,, topological subspace ,, quotient topological space ,, fiber space ,, space attachment ,, mapping cocylinder, mapping cocone ,, mapping cylinder, mapping cone, mapping telescope ,, cell complex, CW-complex ,
References
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