subspace topology



Given a topological space XX in the sense of (Bourbaki 71) (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 initial topology on that map). 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.



Topological subspace inclusions are precisely the regular monomorphisms in Top.

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.


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 is a (closed/open) subspace j:CDj: C \hookrightarrow D.


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.


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.


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 April 1, 2016 06:37:23 by Urs Schreiber (