nLab
subspace topology

Contents

Definition

Given a topological space XX in the sense of Bourbaki (that is, a set XX and a topology τ X\tau_X) and a subset YY of XX, a topology τ Y\tau_Y on YY is said to be the topology induced from τ X\tau_X by the set inclusion YXY \hookrightarrow X if τ Y=τ X pw{Y}{UY|Uτ X}\tau_Y = \tau_X \cap_{pw} \{Y\} \coloneqq \{ U \cap Y | U\in\tau_X\}. In other words, τ Y\tau_Y is the smallest topology on YY such that the inclusion YXY \hookrightarrow X is continuous. 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 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).

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.

Properties

Proposition

Subspace inclusions are precisely the regular monomorphisms in TopTop.

For example, the equalizer of two maps f,g:XYf, g \colon X \stackrel{\to}{\to} Y in TopTop is computed as the equalizer at the underlying-set level, equipped with the subspace topology.

Lemma

The pushout in TopTop of any (closed/open) subspace i:ABi: A \hookrightarrow B along any continuous map f:ACf: A \to C 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.

Revised on November 10, 2013 22:32:54 by Urs Schreiber (89.204.137.233)