nLab
subspace topology

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Definition

(subspace topology)

Let (X,τ X)(X, \tau_X) be a topological space, and let SXS \subset X be a subset of its underlying set. Then the corresponding topological subspace has SS as its underlying set, and its open subsets are those subsets of SS which arise as restrictions of open subsets of XX (i.e. intersections of open subsets of XX with SS):

(U SSopen)(U Xτ X(U S=U XS)). \left( U_S \subset S\,\,\text{open} \right) \,\Leftrightarrow\, \left( \underset{U_X \in \tau_X}{\exists} \left( U_S = U_X \cap S \right) \right) \,.

In other words, τ Y\tau_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 2\mathbb{R}^2:

graphics grabbed from Munkres 75

The pair (Y,τ Y)(Y,\tau_Y) is then said to be a topological subspace of (X,τ X)(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.

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 UiXU \overset{i}{\longrightarrow} X be an injective continuous function between topological spaces. Then this is a subspace inclusion (Def. ) precisely if it satisfies the following universal property:

  • For ZZ any topological space, a function ZfUZ \overset{f}{\longrightarrow} U (of underlying sets) is continuous precisely if the composition ifi \circ f is continuous as a function to XX:

    Z f U if i X \array{ Z &\overset{f}{\longrightarrow}& U \\ &{}_{\mathllap{i \circ f}}\searrow& \Big\downarrow{}^{\mathrlap{i}} \\ && X }

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

SetcoDiscΓTop Set \underoverset {\underset{coDisc}{\longrightarrow}} {\overset{\Gamma}{\longleftarrow}} {\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:TopTop \sharp \;\coloneqq\; coDisc \circ \Gamma \;\colon\; Top \longrightarrow Top

for the induced modal operator on Top (sharp modality). We write

idη id \overset{\eta^{\sharp}}{\longrightarrow} \sharp

for the unit morphism of this adjunction.

Notice that this means that for any topological space ZZ, every function of underlying sets

ZX Z \longrightarrow \sharp X

is continuous functions, hence that continuous functions into X\sharp X are in natural bijection to underlying functions of sets. This is the statement of the adjunction hom-isomorphism:

Hom Top(Z,X)Hom Set(Γ(Z),Γ(X)). Hom_{Top}( Z, \sharp X ) \;\simeq\; Hom_{Set}(\Gamma(Z), \Gamma(X)) \,.
Proposition

Let UiXU \overset{i}{\longrightarrow} 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 η U U i i X η X X \array{ U &\overset{ \eta^\sharp_U }{\longrightarrow}& \sharp U \\ {}^{\mathllap{i}}\Big\downarrow && \Big\downarrow{}^{\mathrlap{\sharp i}} \\ X &\underset{\eta^\sharp_X}{\longrightarrow}& \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

χ U:XProp \chi_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 \stackrel{\to}{\to} Y in Top is computed as the equalizer at 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,

A i B f po g C j D,\array{ A & \stackrel{i}{\hookrightarrow} & B \\ \mathllap{f} \downarrow & po & \downarrow \mathrlap{g} \\ C & \underset{j}{\hookrightarrow} & 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 χ U:C2\chi_U \colon C \to \mathbf{2} and χ V:B2\chi_V \colon B \to \mathbf{2} 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 χ W:D2\chi_W \colon D \to \mathbf{2} which extends the pair of maps χ U,χ V\chi_U, \chi_V. It follows that j 1(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 κ\kappa 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 κ\kappa, then the canonical map F(0)colim iκF(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.

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limitsAAAA\phantom{AAAA}colimits
\, 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

  • 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)

  • Michael Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science 28.6 (2018): 856-941. (arXiv:1509.07584)

Last revised on July 25, 2018 at 11:54:26. See the history of this page for a list of all contributions to it.