Editing Initial topology

{{Short description|Coarsest topology making certain functions continuous}}
In [[general topology]] and related areas of [[mathematics]], the '''initial topology''' (or '''induced topology'''<ref name=Rudin>{{Rudin Walter Functional Analysis| at=sections 3.8 and 3.11}}</ref><ref name=ad>{{cite book |chapter-url=https://link.springer.com/chapter/10.1007%2F978-0-8176-8126-5_3 |last=Adamson |first=Iain T.  |title=A General Topology Workbook |chapter=Induced and Coinduced Topologies |date=1996 |publisher=Birkhäuser, Boston, MA |access-date=July 21, 2020 |quote=...&nbsp;the topology induced on E by the family of mappings&nbsp;... |doi=10.1007/978-0-8176-8126-5_3|pages=23–30 |isbn=978-0-8176-3844-3 }}</ref> or '''strong topology''' or '''limit topology''' or '''projective topology''') on a [[Set (mathematics)|set]] <math>X,</math> with respect to a family of functions on <math>X,</math> is the [[coarsest topology]] on <math>X</math> that makes those functions [[Continuous function (topology)|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 [[Duality (mathematics)|dual]] notion is the [[final topology]], which for a given family of functions mapping to a set <math>Y</math> is the [[finest topology]] on <math>Y</math> that makes those functions continuous.

==Definition==

Given a set <math>X</math> and an [[indexed family]] <math>\left(Y_i\right)_{i \in I}</math> of [[topological space]]s with functions
<math display=block>f_i : X \to Y_i,</math>
the initial topology <math>\tau</math> on <math>X</math> is the [[coarsest topology]] on <math>X</math> such that each
<math display=block>f_i : (X, \tau) \to Y_i</math>
is [[continuous function (topology)|continuous]].

'''Definition in terms of open sets'''

If <math>\left(\tau_i\right)_{i \in I}</math> is a family of topologies <math>X</math> indexed by <math>I \neq \varnothing,</math> then the {{em|[[least upper bound]] topology}} of these topologies is the coarsest topology on <math>X</math> that is finer than each <math>\tau_i.</math> This topology always exists and it is equal to the [[subbase|topology generated by]] <math>{\textstyle \bigcap\limits_{i \in I} \tau_i}.</math>{{sfn|Grothendieck|1973|p=1}}

If for every <math>i \in I,</math> <math>\sigma_i</math> denotes the topology on <math>Y_i,</math> then <math>f_i^{-1}\left(\sigma_i\right) = \left\{f_i^{-1}(V) : V \in \sigma_i\right\}</math> is a topology on <math>X</math>, and the {{em|initial topology of the <math>Y_i</math> by the mappings <math>f_i</math>}} is the least upper bound topology of the <math>I</math>-indexed family of topologies <math>f_i^{-1}\left(\sigma_i\right)</math> (for <math>i \in I</math>).{{sfn|Grothendieck|1973|p=1}} 
Explicitly, the initial topology is the collection of open sets [[subbase|generated]] by all sets of the form <math>f_i^{-1}(U),</math> where <math>U</math> is an [[open set]] in <math>Y_i</math> for some <math>i \in I,</math> under finite intersections and arbitrary unions. 

Sets of the form <math>f_i^{-1}(V)</math> are often called {{em|[[cylinder set]]s}}. If <math>I</math> contains [[Singleton set|exactly one element]], then all the open sets of the initial topology <math>(X, \tau)</math> 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 map]]s.
* 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 form]]s of its [[dual space]].
* Given a [[indexed family|family]] of topologies <math>\left\{\tau_i\right\}</math> on a fixed set <math>X</math> the initial topology on <math>X</math> with respect to the functions <math>\operatorname{id}_i : X \to \left(X, \tau_i\right)</math> is the [[supremum]] (or join) of the topologies <math>\left\{\tau_i\right\}</math> in the [[lattice of topologies]] on <math>X.</math> That is, the initial topology <math>\tau</math> is the topology generated by the [[union (set theory)|union]] of the topologies <math>\left\{\tau_i\right\}.</math>
* A topological space is [[completely regular]] if and only if it has the initial topology with respect to its family of ([[bounded function|bounded]]) real-valued continuous functions.
* Every topological space <math>X</math> has the initial topology with respect to the family of continuous functions from <math>X</math> to the [[Sierpiński space]].

==Properties==

===Characteristic property===

The initial topology on <math>X</math> can be characterized by the following characteristic property:<br>
A function <math>g</math> from some space <math>Z</math> to <math>X</math> is continuous if and only if <math>f_i \circ g</math> is continuous for each <math>i \in I.</math>{{sfn|Grothendieck|1973|p=2}} 

[[Image:InitialTopology-01.png|center|Characteristic property of the initial topology]]

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

A [[Filter (set theory)|filter]] <math>\mathcal{B}</math> on <math>X</math> [[Convergent filter|converges to]] a point <math>x \in X</math> if and only if the [[prefilter]] <math>f_i(\mathcal{B})</math> [[Convergent prefilter|converges to]] <math>f_i(x)</math> for every <math>i \in I.</math>{{sfn|Grothendieck|1973|p=2}}

===Evaluation===

By the universal property of the [[product topology]], we know that any family of continuous maps <math>f_i : X \to Y_i</math> determines a unique continuous map
<math display=block>\begin{alignat}{4}
f :\;&& X &&\;\to    \;& \prod_i Y_i \\[0.3ex]
     && x &&\;\mapsto\;& \left(f_i(x)\right)_{i \in I} \\
\end{alignat}</math>

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 <math>\{f_i : X \to Y_i\}</math> is said to ''[[Separating set|{{visible anchor|separate points}}]]'' in <math>X</math> if for all <math>x \neq y</math> in <math>X</math> there exists some <math>i</math> such that <math>f_i(x) \neq f_i(y).</math> The family <math>\{f_i\}</math> separates points if and only if the associated evaluation map <math>f</math> is [[injective]].

The evaluation map <math>f</math> will be a [[topological embedding]] if and only if <math>X</math> has the initial topology determined by the maps <math>\{f_i\}</math> and this family of maps separates points in <math>X.</math>

===Hausdorffness===

If <math>X</math> has the initial topology induced by <math>\left\{f_i : X \to Y_i\right\}</math> and if every <math>Y_i</math> is Hausdorff, then <math>X</math> is a [[Hausdorff space]] if and only if these maps [[#separate points|separate points]] on <math>X.</math>{{sfn|Grothendieck|1973|p=1}}

===Transitivity of the initial topology===

If <math>X</math> has the initial topology induced by the <math>I</math>-indexed family of mappings <math>\left\{f_i : X \to Y_i\right\}</math> and if for every <math>i \in I,</math> the topology on <math>Y_i</math> is the initial topology induced by some <math>J_i</math>-indexed family of mappings <math>\left\{g_j : Y_i \to Z_j\right\}</math> (as <math>j</math> ranges over <math>J_i</math>), then the initial topology on <math>X</math> induced by <math>\left\{f_i : X \to Y_i\right\}</math> is equal to the initial topology induced by the <math>{\textstyle \bigcup\limits_{i \in I} J_i}</math>-indexed family of mappings <math>\left\{g_j \circ f_i : X \to Z_j\right\}</math> as <math>i</math> ranges over <math>I</math> and <math>j</math> ranges over <math>J_i.</math>{{sfn|Grothendieck|1973|pp=1-2}} 
Several important corollaries of this fact are now given. 

In particular, if <math>S \subseteq X</math> then the subspace topology that <math>S</math> inherits from <math>X</math> is equal to the initial topology induced by the [[inclusion map]] <math>S \to X</math> (defined by <math>s \mapsto s</math>). Consequently, if <math>X</math> has the initial topology induced by <math>\left\{f_i : X \to Y_i\right\}</math> then the subspace topology that <math>S</math> inherits from <math>X</math> is equal to the initial topology induced on <math>S</math> by the restrictions <math>\left\{\left.f_i\right|_S : S \to Y_i\right\}</math> of the <math>f_i</math> to <math>S.</math>{{sfn|Grothendieck|1973|p=2}} 

The [[product topology]] on <math>\prod_i Y_i</math> is equal to the initial topology induced by the canonical projections <math>\operatorname{pr}_i : \left(x_k\right)_{k \in I} \mapsto x_i</math> as <math>i</math> ranges over <math>I.</math>{{sfn|Grothendieck|1973|p=2}}
Consequently, the initial topology on <math>X</math> induced by <math>\left\{f_i : X \to Y_i\right\}</math> is equal to the inverse image of the product topology on <math>\prod_i Y_i</math> by the [[#evaluation map|evaluation map]] <math display=inline>f : X \to \prod_i Y_i\,.</math>{{sfn|Grothendieck|1973|p=2}}  Furthermore, if the maps <math>\left\{f_i\right\}_{i \in I}</math> [[#separate points|separate points]] on <math>X</math> then the evaluation map is a [[homeomorphism]] onto the subspace <math>f(X)</math> of the product space <math>\prod_i Y_i.</math>{{sfn|Grothendieck|1973|p=2}}

===Separating points from closed sets===

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

A family of maps <math>\left\{f_i : X \to Y_i\right\}</math> ''separates points from closed sets'' in <math>X</math> if for all [[closed set]]s <math>A</math> in <math>X</math> and all <math>x \not\in A,</math> there exists some <math>i</math> such that
<math display=block>f_i(x) \notin \operatorname{cl}(f_i(A))</math>
where <math>\operatorname{cl}</math> denotes the [[Closure (topology)|closure operator]].

:'''Theorem'''. A family of continuous maps <math>\left\{f_i : X \to Y_i\right\}</math> separates points from closed sets if and only if the cylinder sets <math>f_i^{-1}(V),</math> for <math>V</math> open in <math>Y_i,</math> form a [[Base (topology)|base for the topology]] on <math>X.</math>

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

If the space <math>X</math> is a [[T0 space|T<sub>0</sub> space]], then any collection of maps <math>\left\{f_i\right\}</math> that separates points from closed sets in <math>X</math> must also separate points. In this case, the evaluation map will be an embedding.

===Initial uniform structure===

{{Main|Uniform space}}

If <math>\left(\mathcal{U}_i\right)_{i \in I}</math> is a family of [[uniform structure]]s on <math>X</math> indexed by <math>I \neq \varnothing,</math> then the {{em|[[least upper bound]] uniform structure}} of <math>\left(\mathcal{U}_i\right)_{i \in I}</math> is the coarsest uniform structure on <math>X</math> that is finer than each <math>\mathcal{U}_i.</math> This uniform always exists and it is equal to the [[Filter (set theory)|filter]] on <math>X \times X</math> generated by the [[filter subbase]] <math>{\textstyle \bigcup\limits_{i \in I} \mathcal{U}_i}.</math>{{sfn|Grothendieck|1973|p=3}} 
If <math>\tau_i</math> is the topology on <math>X</math> induced by the uniform structure <math>\mathcal{U}_i</math> then the topology on <math>X</math> associated with least upper bound uniform structure is equal to the least upper bound topology of <math>\left(\tau_i\right)_{i \in I}.</math>{{sfn|Grothendieck|1973|p=3}} 

Now suppose that <math>\left\{f_i : X \to Y_i\right\}</math> is a family of maps and for every <math>i \in I,</math> let <math>\mathcal{U}_i</math> be a uniform structure on <math>Y_i.</math> Then the {{em|initial uniform structure of the <math>Y_i</math> by the mappings <math>f_i</math>}} is the unique coarsest uniform structure <math>\mathcal{U}</math> on <math>X</math> making all <math>f_i : \left(X, \mathcal{U}\right) \to \left(Y_i, \mathcal{U}_i\right)</math> [[uniformly continuous]].{{sfn|Grothendieck|1973|p=3}} It is equal to the least upper bound uniform structure of the <math>I</math>-indexed family of uniform structures <math>f_i^{-1}\left(\mathcal{U}_i\right)</math> (for <math>i \in I</math>).{{sfn|Grothendieck|1973|p=3}} 
The topology on <math>X</math> induced by <math>\mathcal{U}</math> is the coarsest topology on <math>X</math> such that every <math>f_i : X \to Y_i</math> is continuous.{{sfn|Grothendieck|1973|p=3}} 
The initial uniform structure <math>\mathcal{U}</math> is also equal to the coarsest uniform structure such that the identity mappings <math>\operatorname{id} : \left(X, \mathcal{U}\right) \to \left(X, f_i^{-1}\left(\mathcal{U}_i\right)\right)</math> are uniformly continuous.{{sfn|Grothendieck|1973|p=3}} 

'''Hausdorffness''': The topology on <math>X</math> induced by the initial uniform structure <math>\mathcal{U}</math> is [[Hausdorff space|Hausdorff]] if and only if for whenever <math>x, y \in X</math> are distinct (<math>x \neq y</math>) then there exists some <math>i \in I</math> and some entourage <math>V_i \in \mathcal{U}_i</math> of <math>Y_i</math> such that <math>\left(f_i(x), f_i(y)\right) \not\in V_i.</math>{{sfn|Grothendieck|1973|p=3}} 
Furthermore, if for every index <math>i \in I,</math> the topology on <math>Y_i</math> induced by <math>\mathcal{U}_i</math> is Hausdorff then the topology on <math>X</math> induced by the initial uniform structure <math>\mathcal{U}</math> is Hausdorff if and only if the maps <math>\left\{f_i : X \to Y_i\right\}</math> [[#separate points|separate points]] on <math>X</math>{{sfn|Grothendieck|1973|p=3}} (or equivalently, if and only if the [[#evaluation map|evaluation map]] <math display=inline>f : X \to \prod_i Y_i</math> is injective)

'''Uniform continuity''': If <math>\mathcal{U}</math> is the initial uniform structure induced by the mappings <math>\left\{f_i : X \to Y_i\right\},</math> then a function <math>g</math> from some uniform space <math>Z</math> into <math>(X, \mathcal{U})</math> is [[uniformly continuous]] if and only if <math>f_i \circ g : Z \to Y_i</math> is uniformly continuous for each <math>i \in I.</math>{{sfn|Grothendieck|1973|p=3}} 

'''Cauchy filter''': A [[Filter (set theory)|filter]] <math>\mathcal{B}</math> on <math>X</math> is a [[Cauchy filter]] on <math>(X, \mathcal{U})</math> if and only if <math>f_i\left(\mathcal{B}\right)</math> is a Cauchy prefilter on <math>Y_i</math> for every <math>i \in I.</math>{{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|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 <math>Y</math> be the [[functor]] from a [[discrete category]] <math>J</math> to the [[category of topological spaces]] <math>\mathrm{Top}</math> which maps <math>j\mapsto Y_j</math>. Let <math>U</math> be the usual [[forgetful functor]] from <math>\mathrm{Top}</math> to <math>\mathrm{Set}</math>. The maps <math>f_j : X \to Y_j</math> can then be thought of as a [[cone (category theory)|cone]] from <math>X</math> to <math>UY.</math> That is, <math>(X,f)</math> is an object of <math>\mathrm{Cone}(UY) := (\Delta\downarrow{UY})</math>&mdash;the [[category of cones]] to <math>UY.</math> More precisely, this cone <math>(X,f)</math> defines a <math>U</math>-structured cosink in <math>\mathrm{Set}.</math>

The forgetful functor <math>U : \mathrm{Top} \to \mathrm{Set}</math> induces a functor <math>\bar{U} : \mathrm{Cone}(Y) \to \mathrm{Cone}(UY)</math>. The characteristic property of the initial topology is equivalent to the statement that there exists a [[universal morphism]] from <math>\bar{U}</math> to <math>(X,f);</math> that is, a [[terminal object]] in the category <math>\left(\bar{U}\downarrow(X,f)\right).</math><br/>
Explicitly, this consists of an object <math>I(X,f)</math> in <math>\mathrm{Cone}(Y)</math> together with a morphism <math>\varepsilon : \bar{U} I(X,f) \to (X,f)</math> such that for any object <math>(Z,g)</math> in <math>\mathrm{Cone}(Y)</math> and morphism <math>\varphi : \bar{U}(Z,g) \to (X,f)</math> there exists a unique morphism <math>\zeta : (Z,g) \to I(X,f)</math> such that the following diagram commutes:
[[File:UniversalPropInitialTop.jpg|300px|center]]
The assignment <math>(X,f) \mapsto I(X,f)</math> placing the initial topology on <math>X</math> extends to a functor
<math>I : \mathrm{Cone}(UY) \to \mathrm{Cone}(Y)</math>
which is [[adjoint functor|right adjoint]] to the forgetful functor <math>\bar{U}.</math> In fact, <math>I</math> is a right-inverse to <math>\bar{U}</math>; since <math>\bar{U}I</math> is the identity functor on <math>\mathrm{Cone}(UY).</math>

==See also==

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

==References==

{{reflist}}

==Bibliography==

* {{Bourbaki General Topology Part I Chapters 1-4}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Bourbaki General Topology Part II Chapters 5-10}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Dugundji Topology}} <!--{{sfn|Dugundji|1966|p=}}-->
* {{Grothendieck Topological Vector Spaces}} <!--{{sfn|Grothendieck|1973|p=}}-->
* {{Willard General Topology}} <!--{{sfn|Willard|2004|p=}}-->
* {{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}}

==External links==

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

{{Topology|expanded}}

[[Category:General topology]]