The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology

Two constructions are defined here, one for maps into spaces and one for maps out of spaces. Every further construction on this page is an instance of one of them.

Initial topology. Let $X$ be a set, let $I$ be an index set, let $(Y_i,{\mathcal{T}}_i)$ be a topological space for each $i\in I$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), and let $f_i:X\to Y_i$ be a function for each $i$. The initial topology on $X$ induced by the family $(f_i{)}_{i\in I}$ is

${\mathcal{T}}^{\mathrm{i}\mathrm{n}}:=⟨\{f_i^{-1}[V]:i\in I,V\in {\mathcal{T}}_i\}⟩,$

the topology generated by the preimages of the open sets of the $Y_i$ (Basis and subbasis for a topology, and the topology generated by a family of sets).

This is well posed, and the obligation is discharged by the item cited. For an arbitrary family $\mathcal{S}$ of subsets of $X$, the family $⟨\mathcal{S}⟩$ is a topology on $X$, contains $\mathcal{S}$, and is contained in every topology on $X$ containing $\mathcal{S}$ (Basis and subbasis for a topology, and the topology generated by a family of sets); no further verification is needed here. By A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis a basis for ${\mathcal{T}}^{\mathrm{i}\mathrm{n}}$ is the family of intersections of finitely many sets $f_i^{-1}[V]$, the intersection of none being $X$.

Final topology. Let $X$ be a set, let $(Z_i,{\mathcal{S}}_i)$ be a topological space for each $i\in I$, and let $g_i:Z_i\to X$ be a function for each $i$. The final topology on $X$ induced by the family $(g_i{)}_{i\in I}$ is

${\mathcal{T}}^{\mathrm{f}\mathrm{i}\mathrm{n}}:=\{U\subseteq X:g_i^{-1}[U]\in {\mathcal{S}}_i\text{ for every }i\in I\}\mathrm{.}$

This is a topology, and the verification is carried out here rather than assumed. Preimage commutes with the three set operations named in the axioms: $g_i^{-1}[\mathrm{\varnothing }]=\mathrm{\varnothing }$ and $g_i^{-1}[X]=Z_i$, which gives (T1); $g_i^{-1}[\bigcup \mathcal{U}]=\bigcup \{g_i^{-1}[U]:U\in \mathcal{U}\}$ for every family $\mathcal{U}\subseteq {\mathcal{T}}^{\mathrm{f}\mathrm{i}\mathrm{n}}$, which with (T2) in $Z_i$ gives (T2); and $g_i^{-1}[U\cap V]=g_i^{-1}[U]\cap g_i^{-1}[V]$, which with (T3) in $Z_i$ gives (T3) (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

Two degenerate cases, stated because they are used. For $I=\mathrm{\varnothing }$ the initial topology is $⟨\mathrm{\varnothing }⟩=\{\mathrm{\varnothing },X\}$, the indiscrete topology, and the final topology is $\mathcal{P}(X)$, the discrete topology, the defining condition being vacuous (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Basis and subbasis for a topology, and the topology generated by a family of sets). For a family consisting of a single map the definitions read the same way with the index dropped.

The subspace topology is the model initial topology. Let $(X,\mathcal{T})$ be a space, let $S\subseteq X$ and let $\iota :S\to X$ be the inclusion. Then ${\iota }^{-1}[U]=U\cap S$ for every $U\subseteq X$, so the generating family for the initial topology of the one-element family $(\iota )$ is exactly ${\mathcal{T}}_S=\{U\cap S:U\in \mathcal{T}\}$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). That family is already a topology, so generating adds nothing and the initial topology of $\iota$ is the subspace topology. Everything Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace proves about it is therefore quoted here and not reproved: its closed sets are the traces of the closed sets, a basis of $X$ traces to a basis of $S$, the inclusion is continuous (Continuity of a map of topological spaces at a point and globally, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f(\bar{A})\subseteq \overline{f(A)}$), ${\mathcal{T}}_S$ is the coarsest topology making $\iota$ continuous, and a map $g:Z\to S$ is continuous if and only if $\iota \circ g$ is.

Terminology. A family $(f_i)$ as above is the defining family of the initial topology, and likewise for the final one; both topologies depend on the whole family and not on any one member. The characteristic properties that make these two constructions worth naming are the subject of the next item.

• The direction of the arrows is what distinguishes the two. An initial topology is put on the source of its defining maps and is the coarsest making them continuous; a final topology is put on the target and is the finest making them continuous. Both facts are proved in the next item rather than built into the definitions above.

• Why the initial topology is generated and the final one is not. The preimages $f_i^{-1}[V]$ need not be closed under unions or finite intersections, so a topology has to be generated from them; the final family is closed under both operations already, because preimage commutes with both, so it is a topology as it stands. The asymmetry is a fact about preimages and not a choice of presentation.

• Both constructions are determined by a universal property, so they are unique once that property is stated. That is the content of the next item, and it is what lets the product, the coproduct and the quotient be treated as three instances of two constructions rather than as three separate theories.