initial topology

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

$$f\_i\; :\; X\; \backslash to\; Y\_i,$$

the initial topology $\backslash tau$ on $X$ is the coarsest topology on $X$ such that each

$$f\_i\; :\; (X,\; \backslash tau)\; \backslash to\; Y\_i$$

is continuous.

Definition in terms of open sets

If $\backslash left(\backslash tau\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a family of topologies $X$ indexed by $I\; \backslash neq\; \backslash varnothing,$ then the {{em|least upper bound topology}} of these topologies is the coarsest topology on $X$ that is finer than each $\backslash tau\_i.$ This topology always exists and it is equal to the topology generated by $\backslash bigcup\_\{i\; \backslash in\; I\}\; \backslash tau\_i.${{sfn|Grothendieck|1973|p=1}}

If for every $i\; \backslash in\; I,$ $\backslash sigma\_i$ denotes the topology on $Y\_i,$ then $f\_i^\{-1\}\backslash left(\backslash sigma\_i\backslash right)\; =\; \backslash left\backslash \{f\_i^\{-1\}(V)\; :\; V\; \backslash in\; \backslash sigma\_i\backslash right\backslash \}$ is a topology on $X$, and the {{em|initial topology of the $Y\_i$ by the mappings $f\_i$}} is the least upper bound topology of the $I$-indexed family of topologies $f\_i^\{-1\}\backslash left(\backslash sigma\_i\backslash right)$ (for $i\; \backslash in\; I$).{{sfn|Grothendieck|1973|p=1}}

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 $i\; \backslash in\; I,$ under finite intersections and arbitrary unions.

Sets of the form $f\_i^\{-1\}(V)$ are often called {{em|cylinder sets}}. If $I$ contains exactly one element, then all the open sets of the initial topology $(X,\; \backslash tau)$ are cylinder sets.

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 $\backslash left\backslash \{\backslash tau\_i\backslash right\backslash \}$ on a fixed set $X$ the initial topology on $X$ with respect to the functions $\backslash operatorname\{id\}\_i\; :\; X\; \backslash to\; \backslash left(X,\; \backslash tau\_i\backslash right)$ is the supremum (or join) of the topologies $\backslash left\backslash \{\backslash tau\_i\backslash right\backslash \}$ in the lattice of topologies on $X.$ That is, the initial topology $\backslash tau$ is the topology generated by the union of the topologies $\backslash left\backslash \{\backslash tau\_i\backslash right\backslash \}.$
• 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 $X$ has the initial topology with respect to the family of continuous functions from $X$ to the Sierpiński space.

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 $f\_i\; \backslash circ\; g$ is continuous for each $i\; \backslash in\; I.${{sfn|Grothendieck|1973|p=2}}

Image:InitialTopology-01.png

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

A filter $\backslash mathcal\{B\}$ on $X$ converges to a point $x\; \backslash in\; X$ if and only if the prefilter $f\_i(\backslash mathcal\{B\})$ converges to $f\_i(x)$ for every $i\; \backslash in\; I.${{sfn|Grothendieck|1973|p=2}}

By the universal property of the product topology, we know that any family of continuous maps $f\_i\; :\; X\; \backslash to\; Y\_i$ determines a unique continuous map

$$\backslash begin\{alignat\}\{4\}$$

f :\;&& X &&\;\to \;& \prod_i Y_i \\[0.3ex]

&& x &&\;\mapsto\;& \left(f_i(x)\right)_{i \in I} \\

\end{alignat}

This map is known as the {{visible anchor|evaluation map}}.{{cn|reason=Such a counterintuitive term must be reliably sourced|date=February 2024}}

A family of maps $\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash \}$ is said to Separating set in $X$ if for all $x\; \backslash neq\; y$ in $X$ there exists some $i$ such that $f\_i(x)\; \backslash neq\; f\_i(y).$ The family $\backslash \{f\_i\backslash \}$ 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 $\backslash \{f\_i\backslash \}$ and this family of maps separates points in $X.$

If $X$ has the initial topology induced by $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ and if every $Y\_i$ is Hausdorff, then $X$ is a Hausdorff space if and only if these maps separate points on $X.${{sfn|Grothendieck|1973|p=1}}

If $X$ has the initial topology induced by the $I$-indexed family of mappings $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ and if for every $i\; \backslash in\; I,$ the topology on $Y\_i$ is the initial topology induced by some $J\_i$-indexed family of mappings $\backslash left\backslash \{g\_j\; :\; Y\_i\; \backslash to\; Z\_j\backslash right\backslash \}$ (as $j$ ranges over $J\_i$), then the initial topology on $X$ induced by $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ is equal to the initial topology induced by the $\{\backslash textstyle\; \backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\; J\_i\}$-indexed family of mappings $\backslash left\backslash \{g\_j\; \backslash circ\; f\_i\; :\; X\; \backslash to\; Z\_j\backslash right\backslash \}$ as $i$ ranges over $I$ and $j$ ranges over $J\_i.${{sfn|Grothendieck|1973|pp=1-2}}

Several important corollaries of this fact are now given.

In particular, if $S\; \backslash subseteq\; X$ then the subspace topology that $S$ inherits from $X$ is equal to the initial topology induced by the inclusion map $S\; \backslash to\; X$ (defined by $s\; \backslash mapsto\; s$). Consequently, if $X$ has the initial topology induced by $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ then the subspace topology that $S$ inherits from $X$ is equal to the initial topology induced on $S$ by the restrictions $\backslash left\backslash \{\backslash left.f\_i\backslash right|\_S\; :\; S\; \backslash to\; Y\_i\backslash right\backslash \}$ of the $f\_i$ to $S.${{sfn|Grothendieck|1973|p=2}}

The product topology on $\backslash prod\_i\; Y\_i$ is equal to the initial topology induced by the canonical projections $\backslash operatorname\{pr\}\_i\; :\; \backslash left(x\_k\backslash right)\_\{k\; \backslash in\; I\}\; \backslash mapsto\; x\_i$ as $i$ ranges over $I.${{sfn|Grothendieck|1973|p=2}}

Consequently, the initial topology on $X$ induced by $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ is equal to the inverse image of the product topology on $\backslash prod\_i\; Y\_i$ by the evaluation map $f\; :\; X\; \backslash to\; \backslash prod\_i\; Y\_i\backslash ,.${{sfn|Grothendieck|1973|p=2}} Furthermore, if the maps $\backslash left\backslash \{f\_i\backslash right\backslash \}\_\{i\; \backslash in\; I\}$ separate points on $X$ then the evaluation map is a homeomorphism onto the subspace $f(X)$ of the product space $\backslash prod\_i\; Y\_i.${{sfn|Grothendieck|1973|p=2}}

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 $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ separates points from closed sets in $X$ if for all closed sets $A$ in $X$ and all $x\; \backslash not\backslash in\; A,$ there exists some $i$ such that

$$f\_i(x)\; \backslash notin\; \backslash operatorname\{cl\}(f\_i(A))$$

where $\backslash operatorname\{cl\}$ denotes the closure operator.

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

It follows that whenever $\backslash left\backslash \{f\_i\backslash right\backslash \}$ separates points from closed sets, the space $X$ has the initial topology induced by the maps $\backslash left\backslash \{f\_i\backslash right\backslash \}.$ 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 $\backslash left\backslash \{f\_i\backslash right\backslash \}$ that separates points from closed sets in $X$ must also separate points. In this case, the evaluation map will be an embedding.

{{Main|Uniform space}}

If $\backslash left(\backslash mathcal\{U\}\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a family of uniform structures on $X$ indexed by $I\; \backslash neq\; \backslash varnothing,$ then the {{em|least upper bound uniform structure}} of $\backslash left(\backslash mathcal\{U\}\_i\backslash right)\_\{i\; \backslash in\; I\}$ is the coarsest uniform structure on $X$ that is finer than each $\backslash mathcal\{U\}\_i.$ This uniform always exists and it is equal to the filter on $X\; \backslash times\; X$ generated by the filter subbase $\{\backslash textstyle\; \backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\; \backslash mathcal\{U\}\_i\}.${{sfn|Grothendieck|1973|p=3}}

If $\backslash tau\_i$ is the topology on $X$ induced by the uniform structure $\backslash mathcal\{U\}\_i$ then the topology on $X$ associated with least upper bound uniform structure is equal to the least upper bound topology of $\backslash left(\backslash tau\_i\backslash right)\_\{i\; \backslash in\; I\}.${{sfn|Grothendieck|1973|p=3}}

Now suppose that $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ is a family of maps and for every $i\; \backslash in\; I,$ let $\backslash mathcal\{U\}\_i$ be a uniform structure on $Y\_i.$ Then the {{em|initial uniform structure of the $Y\_i$ by the mappings $f\_i$}} is the unique coarsest uniform structure $\backslash mathcal\{U\}$ on $X$ making all $f\_i\; :\; \backslash left(X,\; \backslash mathcal\{U\}\backslash right)\; \backslash to\; \backslash left(Y\_i,\; \backslash mathcal\{U\}\_i\backslash right)$ uniformly continuous.{{sfn|Grothendieck|1973|p=3}} It is equal to the least upper bound uniform structure of the $I$-indexed family of uniform structures $f\_i^\{-1\}\backslash left(\backslash mathcal\{U\}\_i\backslash right)$ (for $i\; \backslash in\; I$).{{sfn|Grothendieck|1973|p=3}}

The topology on $X$ induced by $\backslash mathcal\{U\}$ is the coarsest topology on $X$ such that every $f\_i\; :\; X\; \backslash to\; Y\_i$ is continuous.{{sfn|Grothendieck|1973|p=3}}

The initial uniform structure $\backslash mathcal\{U\}$ is also equal to the coarsest uniform structure such that the identity mappings $\backslash operatorname\{id\}\; :\; \backslash left(X,\; \backslash mathcal\{U\}\backslash right)\; \backslash to\; \backslash left(X,\; f\_i^\{-1\}\backslash left(\backslash mathcal\{U\}\_i\backslash right)\backslash right)$ are uniformly continuous.{{sfn|Grothendieck|1973|p=3}}

Hausdorffness: The topology on $X$ induced by the initial uniform structure $\backslash mathcal\{U\}$ is Hausdorff if and only if for whenever $x,\; y\; \backslash in\; X$ are distinct ($x\; \backslash neq\; y$) then there exists some $i\; \backslash in\; I$ and some entourage $V\_i\; \backslash in\; \backslash mathcal\{U\}\_i$ of $Y\_i$ such that $\backslash left(f\_i(x),\; f\_i(y)\backslash right)\; \backslash not\backslash in\; V\_i.${{sfn|Grothendieck|1973|p=3}}

Furthermore, if for every index $i\; \backslash in\; I,$ the topology on $Y\_i$ induced by $\backslash mathcal\{U\}\_i$ is Hausdorff then the topology on $X$ induced by the initial uniform structure $\backslash mathcal\{U\}$ is Hausdorff if and only if the maps $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \}$ separate points on $X${{sfn|Grothendieck|1973|p=3}} (or equivalently, if and only if the evaluation map $f\; :\; X\; \backslash to\; \backslash prod\_i\; Y\_i$ is injective)

Uniform continuity: If $\backslash mathcal\{U\}$ is the initial uniform structure induced by the mappings $\backslash left\backslash \{f\_i\; :\; X\; \backslash to\; Y\_i\backslash right\backslash \},$ then a function $g$ from some uniform space $Z$ into $(X,\; \backslash mathcal\{U\})$ is uniformly continuous if and only if $f\_i\; \backslash circ\; g\; :\; Z\; \backslash to\; Y\_i$ is uniformly continuous for each $i\; \backslash in\; I.${{sfn|Grothendieck|1973|p=3}}

Cauchy filter: A filter $\backslash mathcal\{B\}$ on $X$ is a Cauchy filter on $(X,\; \backslash mathcal\{U\})$ if and only if $f\_i\backslash left(\backslash mathcal\{B\}\backslash right)$ is a Cauchy prefilter on $Y\_i$ for every $i\; \backslash in\; I.${{sfn|Grothendieck|1973|p=3}}

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.

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 $\backslash mathrm\{Top\}$ which maps $j\backslash mapsto\; Y\_j$. Let $U$ be the usual forgetful functor from $\backslash mathrm\{Top\}$ to $\backslash mathrm\{Set\}$. The maps $f\_j\; :\; X\; \backslash to\; Y\_j$ can then be thought of as a cone from $X$ to $UY.$ That is, $(X,f)$ is an object of $\backslash mathrm\{Cone\}(UY)\; :=\; (\backslash Delta\backslash downarrow\{UY\})$—the category of cones to $UY.$ More precisely, this cone $(X,f)$ defines a $U$-structured cosink in $\backslash mathrm\{Set\}.$

The forgetful functor $U\; :\; \backslash mathrm\{Top\}\; \backslash to\; \backslash mathrm\{Set\}$ induces a functor $\backslash bar\{U\}\; :\; \backslash mathrm\{Cone\}(Y)\; \backslash to\; \backslash mathrm\{Cone\}(UY)$. The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from $\backslash bar\{U\}$ to $(X,f);$ that is, a terminal object in the category $\backslash left(\backslash bar\{U\}\backslash downarrow(X,f)\backslash right).$

Explicitly, this consists of an object $I(X,f)$ in $\backslash mathrm\{Cone\}(Y)$ together with a morphism $\backslash varepsilon\; :\; \backslash bar\{U\}\; I(X,f)\; \backslash to\; (X,f)$ such that for any object $(Z,g)$ in $\backslash mathrm\{Cone\}(Y)$ and morphism $\backslash varphi\; :\; \backslash bar\{U\}(Z,g)\; \backslash to\; (X,f)$ there exists a unique morphism $\backslash zeta\; :\; (Z,g)\; \backslash to\; I(X,f)$ such that the following diagram commutes:

File:UniversalPropInitialTop.jpg

The assignment $(X,f)\; \backslash mapsto\; I(X,f)$ placing the initial topology on $X$ extends to a functor

$I\; :\; \backslash mathrm\{Cone\}(UY)\; \backslash to\; \backslash mathrm\{Cone\}(Y)$

which is right adjoint to the forgetful functor $\backslash bar\{U\}.$ In fact, $I$ is a right-inverse to $\backslash bar\{U\}$; since $\backslash bar\{U\}I$ is the identity functor on $\backslash mathrm\{Cone\}(UY).$

• {{annotated link|Final topology}}
• {{annotated link|Product topology}}
• {{annotated link|Quotient space (topology)}}
• {{annotated link|Subspace topology}}

{{reflist}}

• {{Bourbaki General Topology Part I Chapters 1-4}}
• {{Bourbaki General Topology Part II Chapters 5-10}}
• {{Dugundji Topology}}
• {{Grothendieck Topological Vector Spaces}}
• {{Willard General Topology}}
• {{cite book | last=Willard | first=Stephen | title=General Topology | url=https://archive.org/details/generaltopology00will_0 | url-access=registration | publisher=Addison-Wesley | location=Reading, Massachusetts | year=1970 | isbn=0-486-43479-6}}

• {{PlanetMath |urlname=initialtopology |title=Initial topology}}
• {{PlanetMath |urlname=producttopologyandsubspacetopology |title=Product topology and subspace topology}}

{{Topology|expanded}}

Category:General topology