In general topology and related areas of mathematics, the initial topology (or induced topology^[1][2] or weak topology or limit topology or projective topology) on a set ${\displaystyle X,}$ with respect to a family of functions on ${\displaystyle X,}$ is the coarsest topology on ${\displaystyle X}$ that makes those functions continuous.

The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.

The dual notion is the final topology, which for a given family of functions mapping to a set ${\displaystyle X}$ is the finest topology on ${\displaystyle X}$ that makes those functions continuous.

Definition

Given a set ${\displaystyle X}$ and an indexed family ${\displaystyle \left(Y_{i}\right)_{i\in I))$ of topological spaces with functions

$${\displaystyle f_{i}:X\to Y_{i},}$$

${\displaystyle \tau }$

${\displaystyle X}$

${\displaystyle X}$

$${\displaystyle f_{i}:(X,\tau )\to Y_{i))$$

Definition in terms of open sets

If ${\displaystyle \left(\tau _{i}\right)_{i\in I))$ is a family of topologies ${\displaystyle X}$ indexed by ${\displaystyle I\neq \varnothing ,}$ then the least upper bound topology of these topologies is the coarsest topology on ${\displaystyle X}$ that is finer than each ${\displaystyle \tau _{i}.}$ This topology always exists and it is equal to the topology generated by ${\displaystyle {\textstyle \bigcup \limits _{i\in I}\tau _{i)).}$^[3]

If for every ${\displaystyle i\in I,}$ ${\displaystyle \sigma _{i))$ denotes the topology on ${\displaystyle Y_{i},}$ then ${\displaystyle f_{i}^{-1}\left(\sigma _{i}\right)=\left\{f_{i}^{-1}(V):V\in \sigma _{i}\right\))$ is a topology on ${\displaystyle X}$, and the initial topology of the ${\displaystyle Y_{i))$ by the mappings ${\displaystyle f_{i))$ is the least upper bound topology of the ${\displaystyle I}$-indexed family of topologies ${\displaystyle f_{i}^{-1}\left(\sigma _{i}\right)}$ (for ${\displaystyle i\in I}$).^[3] Explicitly, the initial topology is the collection of open sets generated by all sets of the form ${\displaystyle f_{i}^{-1}(U),}$ where ${\displaystyle U}$ is an open set in ${\displaystyle Y_{i))$ for some ${\displaystyle i\in I,}$ under finite intersections and arbitrary unions.

Sets of the form ${\displaystyle f_{i}^{-1}(V)}$ are often called cylinder sets. If ${\displaystyle I}$ contains exactly one element, then all the open sets of the initial topology ${\displaystyle (X,\tau )}$ are cylinder sets.

Examples

Several topological constructions can be regarded as special cases of the initial topology.

• The subspace topology is the initial topology on the subspace with respect to the inclusion map.
• The product topology is the initial topology with respect to the family of projection maps.
• The inverse limit of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
• The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space.
• Given a family of topologies ${\displaystyle \left\{\tau _{i}\right\))$ on a fixed set ${\displaystyle X}$ the initial topology on ${\displaystyle X}$ with respect to the functions ${\displaystyle \operatorname {id} _{i}:X\to \left(X,\tau _{i}\right)}$ is the supremum (or join) of the topologies ${\displaystyle \left\{\tau _{i}\right\))$ in the lattice of topologies on ${\displaystyle X.}$ That is, the initial topology ${\displaystyle \tau }$ is the topology generated by the union of the topologies ${\displaystyle \left\{\tau _{i}\right\}.}$
• A topological space is completely regular if and only if it has the initial topology with respect to its family of (bounded) real-valued continuous functions.
• Every topological space ${\displaystyle X}$ has the initial topology with respect to the family of continuous functions from ${\displaystyle X}$ to the Sierpiński space.

Properties

Characteristic property

The initial topology on ${\displaystyle X}$ can be characterized by the following characteristic property:
A function ${\displaystyle g}$ from some space ${\displaystyle Z}$ to ${\displaystyle X}$ is continuous if and only if ${\displaystyle f_{i}\circ g}$ is continuous for each ${\displaystyle i\in I.}$^[4]

Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.

A filter ${\displaystyle {\mathcal {B))}$ on ${\displaystyle X}$ converges to a point ${\displaystyle x\in X}$ if and only if the prefilter ${\displaystyle f_{i}({\mathcal {B)))}$ converges to ${\displaystyle f_{i}(x)}$ for every ${\displaystyle i\in I.}$^[4]

Evaluation

By the universal property of the product topology, we know that any family of continuous maps ${\displaystyle f_{i}:X\to Y_{i))$ determines a unique continuous map

$${\displaystyle {\begin{alignedat}{4}f:\;&&X&&\;\to \;&\prod _{i}Y_{i}\\[0.3ex]&&x&&\;\mapsto \;&\left(f_{i}(x)\right)_{i\in I}\\\end{alignedat))}$$

This map is known as the evaluation map.^{[citation needed]}

A family of maps ${\displaystyle \{f_{i}:X\to Y_{i}\))$ is said to separate points in ${\displaystyle X}$ if for all ${\displaystyle x\neq y}$ in ${\displaystyle X}$ there exists some ${\displaystyle i}$ such that ${\displaystyle f_{i}(x)\neq f_{i}(y).}$ The family ${\displaystyle \{f_{i}\))$ separates points if and only if the associated evaluation map ${\displaystyle f}$ is injective.

The evaluation map ${\displaystyle f}$ will be a topological embedding if and only if ${\displaystyle X}$ has the initial topology determined by the maps ${\displaystyle \{f_{i}\))$ and this family of maps separates points in ${\displaystyle X.}$

Hausdorffness

If ${\displaystyle X}$ has the initial topology induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ and if every ${\displaystyle Y_{i))$ is Hausdorff, then ${\displaystyle X}$ is a Hausdorff space if and only if these maps separate points on ${\displaystyle X.}$^[3]

Transitivity of the initial topology

If ${\displaystyle X}$ has the initial topology induced by the ${\displaystyle I}$-indexed family of mappings ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ and if for every ${\displaystyle i\in I,}$ the topology on ${\displaystyle Y_{i))$ is the initial topology induced by some ${\displaystyle J_{i))$-indexed family of mappings ${\displaystyle \left\{g_{j}:Y_{i}\to Z_{j}\right\))$ (as ${\displaystyle j}$ ranges over ${\displaystyle J_{i))$), then the initial topology on ${\displaystyle X}$ induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ is equal to the initial topology induced by the ${\displaystyle {\textstyle \bigcup \limits _{i\in I}J_{i))}$-indexed family of mappings ${\displaystyle \left\{g_{j}\circ f_{i}:X\to Z_{j}\right\))$ as ${\displaystyle i}$ ranges over ${\displaystyle I}$ and ${\displaystyle j}$ ranges over ${\displaystyle J_{i}.}$^[5] Several important corollaries of this fact are now given.

In particular, if ${\displaystyle S\subseteq X}$ then the subspace topology that ${\displaystyle S}$ inherits from ${\displaystyle X}$ is equal to the initial topology induced by the inclusion map ${\displaystyle S\to X}$ (defined by ${\displaystyle s\mapsto s}$). Consequently, if ${\displaystyle X}$ has the initial topology induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ then the subspace topology that ${\displaystyle S}$ inherits from ${\displaystyle X}$ is equal to the initial topology induced on ${\displaystyle S}$ by the restrictions ${\displaystyle \left\{\left.f_{i}\right|_{S}:S\to Y_{i}\right\))$ of the ${\displaystyle f_{i))$ to ${\displaystyle S.}$^[4]

The product topology on ${\displaystyle \prod _{i}Y_{i))$ is equal to the initial topology induced by the canonical projections ${\displaystyle \operatorname {pr} _{i}:\left(x_{k}\right)_{k\in I}\mapsto x_{i))$ as ${\displaystyle i}$ ranges over ${\displaystyle I.}$^[4] Consequently, the initial topology on ${\displaystyle X}$ induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ is equal to the inverse image of the product topology on ${\displaystyle \prod _{i}Y_{i))$ by the evaluation map ${\textstyle f:X\to \prod _{i}Y_{i}\,.}$^[4] Furthermore, if the maps ${\displaystyle \left\{f_{i}\right\}_{i\in I))$ separate points on ${\displaystyle X}$ then the evaluation map is a homeomorphism onto the subspace ${\displaystyle f(X)}$ of the product space ${\displaystyle \prod _{i}Y_{i}.}$^[4]

Separating points from closed sets

If a space ${\displaystyle X}$ comes equipped with a topology, it is often useful to know whether or not the topology on ${\displaystyle X}$ is the initial topology induced by some family of maps on ${\displaystyle X.}$ This section gives a sufficient (but not necessary) condition.

A family of maps ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ separates points from closed sets in ${\displaystyle X}$ if for all closed sets ${\displaystyle A}$ in ${\displaystyle X}$ and all ${\displaystyle x\not \in A,}$ there exists some ${\displaystyle i}$ such that

$${\displaystyle f_{i}(x)\notin \operatorname {cl} (f_{i}(A))}$$

${\displaystyle \operatorname {cl} }$

Theorem. A family of continuous maps ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ separates points from closed sets if and only if the cylinder sets ${\displaystyle f_{i}^{-1}(V),}$ for ${\displaystyle V}$ open in ${\displaystyle Y_{i},}$ form a base for the topology on ${\displaystyle X.}$

It follows that whenever ${\displaystyle \left\{f_{i}\right\))$ separates points from closed sets, the space ${\displaystyle X}$ has the initial topology induced by the maps ${\displaystyle \left\{f_{i}\right\}.}$ The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.

If the space ${\displaystyle X}$ is a T₀ space, then any collection of maps ${\displaystyle \left\{f_{i}\right\))$ that separates points from closed sets in ${\displaystyle X}$ must also separate points. In this case, the evaluation map will be an embedding.

Initial uniform structure

If ${\displaystyle \left({\mathcal {U))_{i}\right)_{i\in I))$ is a family of uniform structures on ${\displaystyle X}$ indexed by ${\displaystyle I\neq \varnothing ,}$ then the least upper bound uniform structure of ${\displaystyle \left({\mathcal {U))_{i}\right)_{i\in I))$ is the coarsest uniform structure on ${\displaystyle X}$ that is finer than each ${\displaystyle {\mathcal {U))_{i}.}$ This uniform always exists and it is equal to the filter on ${\displaystyle X\times X}$ generated by the filter subbase ${\displaystyle {\textstyle \bigcup \limits _{i\in I}{\mathcal {U))_{i)).}$^[6] If ${\displaystyle \tau _{i))$ is the topology on ${\displaystyle X}$ induced by the uniform structure ${\displaystyle {\mathcal {U))_{i))$ then the topology on ${\displaystyle X}$ associated with least upper bound uniform structure is equal to the least upper bound topology of ${\displaystyle \left(\tau _{i}\right)_{i\in I}.}$^[6]

Now suppose that ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ is a family of maps and for every ${\displaystyle i\in I,}$ let ${\displaystyle {\mathcal {U))_{i))$ be a uniform structure on ${\displaystyle Y_{i}.}$ Then the initial uniform structure of the ${\displaystyle Y_{i))$ by the mappings ${\displaystyle f_{i))$ is the unique coarsest uniform structure ${\displaystyle {\mathcal {U))}$ on ${\displaystyle X}$ making all ${\displaystyle f_{i}:\left(X,{\mathcal {U))\right)\to \left(Y_{i},{\mathcal {U))_{i}\right)}$ uniformly continuous.^[6] It is equal to the least upper bound uniform structure of the ${\displaystyle I}$-indexed family of uniform structures ${\displaystyle f_{i}^{-1}\left({\mathcal {U))_{i}\right)}$ (for ${\displaystyle i\in I}$).^[6] The topology on ${\displaystyle X}$ induced by ${\displaystyle {\mathcal {U))}$ is the coarsest topology on ${\displaystyle X}$ such that every ${\displaystyle f_{i}:X\to Y_{i))$ is continuous.^[6] The initial uniform structure ${\displaystyle {\mathcal {U))}$ is also equal to the coarsest uniform structure such that the identity mappings ${\displaystyle \operatorname {id} :\left(X,{\mathcal {U))\right)\to \left(X,f_{i}^{-1}\left({\mathcal {U))_{i}\right)\right)}$ are uniformly continuous.^[6]

Hausdorffness: The topology on ${\displaystyle X}$ induced by the initial uniform structure ${\displaystyle {\mathcal {U))}$ is Hausdorff if and only if for whenever ${\displaystyle x,y\in X}$ are distinct (${\displaystyle x\neq y}$) then there exists some ${\displaystyle i\in I}$ and some entourage ${\displaystyle V_{i}\in {\mathcal {U))_{i))$ of ${\displaystyle Y_{i))$ such that ${\displaystyle \left(f_{i}(x),f_{i}(y)\right)\not \in V_{i}.}$^[6] Furthermore, if for every index ${\displaystyle i\in I,}$ the topology on ${\displaystyle Y_{i))$ induced by ${\displaystyle {\mathcal {U))_{i))$ is Hausdorff then the topology on ${\displaystyle X}$ induced by the initial uniform structure ${\displaystyle {\mathcal {U))}$ is Hausdorff if and only if the maps ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\))$ separate points on ${\displaystyle X}$^[6] (or equivalently, if and only if the evaluation map ${\textstyle f:X\to \prod _{i}Y_{i))$ is injective)

Uniform continuity: If ${\displaystyle {\mathcal {U))}$ is the initial uniform structure induced by the mappings ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\},}$ then a function ${\displaystyle g}$ from some uniform space ${\displaystyle Z}$ into ${\displaystyle (X,{\mathcal {U)))}$ is uniformly continuous if and only if ${\displaystyle f_{i}\circ g:Z\to Y_{i))$ is uniformly continuous for each ${\displaystyle i\in I.}$^[6]

Cauchy filter: A filter ${\displaystyle {\mathcal {B))}$ on ${\displaystyle X}$ is a Cauchy filter on ${\displaystyle (X,{\mathcal {U)))}$ if and only if ${\displaystyle f_{i}\left({\mathcal {B))\right)}$ is a Cauchy prefilter on ${\displaystyle Y_{i))$ for every ${\displaystyle i\in I.}$^[6]

Transitivity of the initial uniform structure: If the word "topology" is replaced with "uniform structure" in the statement of "transitivity of the initial topology" given above, then the resulting statement will also be true.

Categorical description

In the language of category theory, the initial topology construction can be described as follows. Let ${\displaystyle Y}$ be the functor from a discrete category ${\displaystyle J}$ to the category of topological spaces ${\displaystyle \mathrm {Top} }$ which maps ${\displaystyle j\mapsto Y_{j))$. Let ${\displaystyle U}$ be the usual forgetful functor from ${\displaystyle \mathrm {Top} }$ to ${\displaystyle \mathrm {Set} }$. The maps ${\displaystyle f_{j}:X\to Y_{j))$ can then be thought of as a cone from ${\displaystyle X}$ to ${\displaystyle UY.}$ That is, ${\displaystyle (X,f)}$ is an object of ${\displaystyle \mathrm {Cone} (UY):=(\Delta \downarrow {UY})}$—the category of cones to ${\displaystyle UY.}$ More precisely, this cone ${\displaystyle (X,f)}$ defines a ${\displaystyle U}$-structured cosink in ${\displaystyle \mathrm {Set} .}$

The forgetful functor ${\displaystyle U:\mathrm {Top} \to \mathrm {Set} }$ induces a functor ${\displaystyle {\bar {U)):\mathrm {Cone} (Y)\to \mathrm {Cone} (UY)}$. The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from ${\displaystyle {\bar {U))}$ to ${\displaystyle (X,f);}$ that is, a terminal object in the category ${\displaystyle \left({\bar {U))\downarrow (X,f)\right).}$
Explicitly, this consists of an object ${\displaystyle I(X,f)}$ in ${\displaystyle \mathrm {Cone} (Y)}$ together with a morphism ${\displaystyle \varepsilon :{\bar {U))I(X,f)\to (X,f)}$ such that for any object ${\displaystyle (Z,g)}$ in ${\displaystyle \mathrm {Cone} (Y)}$ and morphism ${\displaystyle \varphi :{\bar {U))(Z,g)\to (X,f)}$ there exists a unique morphism ${\displaystyle \zeta :(Z,g)\to I(X,f)}$ such that the following diagram commutes:

The assignment ${\displaystyle (X,f)\mapsto I(X,f)}$ placing the initial topology on ${\displaystyle X}$ extends to a functor ${\displaystyle I:\mathrm {Cone} (UY)\to \mathrm {Cone} (Y)}$ which is right adjoint to the forgetful functor ${\displaystyle {\bar {U)).}$ In fact, ${\displaystyle I}$ is a right-inverse to ${\displaystyle {\bar {U))}$; since ${\displaystyle {\bar {U))I}$ is the identity functor on ${\displaystyle \mathrm {Cone} (UY).}$