Given a topological space in the sense of (Bourbaki 71) (that is, a set and a topology ) and a subset of , a topology on is said to be the topology induced from by the set inclusion if . In other words, is the smallest topology on such that the inclusion is continuous (the initial topology on that map). The pair is then said to be a topological subspace of . The induced topology is for that reason sometimes called the subspace topology on .
A property of topological spaces is said to be hereditary if its satisfaction for a topological space implies its satisfaction for all topological subspaces of .
Since is faithful, we have that monos are reflected by ; also monos and pushouts are preserved by since has both a left adjoint and a right adjoint. In , the pushout of a mono along any map is a mono, so we conclude is monic in . Furthermore, such a pushout diagram in is also a pullback, so that we have the Beck-Chevalley equality (where is the direct image map between power sets, and is the inverse image map).
To prove that is a subspace, let be any open set. Then there exists open such that because is a subspace inclusion. If and 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 which extends the pair of maps . It follows that , so that is a subspace inclusion.
If moreover is an open inclusion, then for any open we have that (since is monic) and (by Beck-Chevalley) is open in . By the definition of the topology on , it follows that is open, so that is an open inclusion. The same proof, replacing the word “open” with the word “closed” throughout, shows that the pushout of a closed inclusion is a closed inclusion .
A similar (but even simpler) line of argument establishes the following result.
Let be an ordinal, viewed as a preorder category, and let be a functor that preserves directed colimits. Then if is a (closed/open) subspace inclusion for each morphism of , then the canonical map 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 (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 ) is a strong topology (or final structure for a sink); an example is the quotient topology on a quotient space.