# Initial topology

In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or projective topology) on a set $X$ , with respect to a family of functions on $X$ , is the coarsest topology on 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 $X$ is the finest topology on $X$ that makes those functions continuous.

## Definition

Given a set X and an indexed family (Yi)iI of topological spaces with functions

$f_{i}:X\to Y_{i},$ the initial topology $\tau$ on $X$ is the coarsest topology on X such that each

$f_{i}:(X,\tau )\to Y_{i}$ is continuous.

Explicitly, the initial topology is the collection of open sets generated by all sets of the form $f_{i}^{-1}(U)$ , where $U$ is an open set in $Y_{i}$ for some i I, under finite intersections and arbitrary unions. The sets $f_{i}^{-1}(U)$ are often called cylinder sets. If I contains exactly one element, all the open sets of $(X,\tau )$ are cylinder sets.

## Examples

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

## Properties

### Characteristic property

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

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

### Evaluation

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

$f\colon X\to \prod _{i}Y_{i}\,.$ This map is known as the evaluation map.

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

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

### Separating points from closed sets

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

A family of maps {fi: X Yi} separates points from closed sets in X if for all closed sets A in X and all x not in A, there exists some i such that

$f_{i}(x)\notin \operatorname {cl} (f_{i}(A))$ where cl denotes the closure operator.

Theorem. A family of continuous maps {fi: X Yi} separates points from closed sets if and only if the cylinder sets $f_{i}^{-1}(U)$ , for U open in Yi, form a base for the topology on X.

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

If the space X is a T0 space, then any collection of maps {fi} that separates points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.

## Categorical description

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

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

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