Final topology

Given a set ${\displaystyle X}$  and an ${\displaystyle I}$ -indexed family of topological spaces ${\displaystyle \left(Y_{i},\upsilon _{i}\right)}$  with associated functions $${\displaystyle f_{i}:Y_{i}\to X,}$$  the final topology on ${\displaystyle X}$  induced by the family of functions ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\}}$  is the finest topology ${\displaystyle \tau _{\mathcal {F}}}$  on ${\displaystyle X}$  such that $${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$$ 

is continuous for each ${\displaystyle i\in I}$ .

Explicitly, the final topology may be described as follows:

a subset ${\displaystyle U}$  of ${\displaystyle X}$  is open in the final topology ${\displaystyle \left(X,\tau _{\mathcal {F}}\right)}$  (that is, ${\displaystyle U\in \tau _{\mathcal {F}}}$ ) if and only if ${\displaystyle f_{i}^{-1}(U)}$  is open in ${\displaystyle \left(Y_{i},\upsilon _{i}\right)}$  for each ${\displaystyle i\in I}$ .

The closed subsets have an analogous characterization:

a subset ${\displaystyle C}$  of ${\displaystyle X}$  is closed in the final topology ${\displaystyle \left(X,\tau _{\mathcal {F}}\right)}$  if and only if ${\displaystyle f_{i}^{-1}(C)}$  is closed in ${\displaystyle \left(Y_{i},\upsilon _{i}\right)}$  for each ${\displaystyle i\in I}$ .

The family ${\displaystyle {\mathcal {F}}}$  of functions that induces the final topology on ${\displaystyle X}$  is usually a set of functions. But the same construction can be performed if ${\displaystyle {\mathcal {F}}}$  is a proper class of functions, and the result is still well-defined in Zermelo–Fraenkel set theory. In that case there is always a subfamily ${\displaystyle {\mathcal {G}}}$  of ${\displaystyle {\mathcal {F}}}$  with ${\displaystyle {\mathcal {G}}}$  a set, such that the final topologies on ${\displaystyle X}$  induced by ${\displaystyle {\mathcal {F}}}$  and by ${\displaystyle {\mathcal {G}}}$  coincide. For more on this, see for example the discussion here.^[4] As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class of functions.^[5]

The important special case where the family of maps ${\displaystyle {\mathcal {F}}}$  consists of a single surjective map can be completely characterized using the notion of quotient map. A surjective function ${\displaystyle f:(Y,\upsilon )\to \left(X,\tau \right)}$  between topological spaces is a quotient map if and only if the topology ${\displaystyle \tau }$  on ${\displaystyle X}$  coincides with the final topology ${\displaystyle \tau _{\mathcal {F}}}$  induced by the family ${\displaystyle {\mathcal {F}}=\{f\}}$ . In particular: the quotient topology is the final topology on the quotient space induced by the quotient map.

The final topology on a set ${\displaystyle X}$  induced by a family of ${\displaystyle X}$ -valued maps can be viewed as a far reaching generalization of the quotient topology, where multiple maps may be used instead of just one and where these maps are not required to be surjections.

Given topological spaces ${\displaystyle X_{i}}$ , the disjoint union topology on the disjoint union ${\displaystyle \coprod _{i}X_{i}}$  is the final topology on the disjoint union induced by the natural injections.

Given a family of topologies ${\displaystyle \left(\tau _{i}\right)_{i\in I}}$  on a fixed set ${\displaystyle X,}$  the final topology on ${\displaystyle X}$  with respect to the identity maps ${\displaystyle \operatorname {id} _{\tau _{i}}:\left(X,\tau _{i}\right)\to X}$  as ${\displaystyle i}$  ranges over ${\displaystyle I,}$  call it ${\displaystyle \tau ,}$  is the infimum (or meet) of these topologies ${\displaystyle \left(\tau _{i}\right)_{i\in I}}$  in the lattice of topologies on ${\displaystyle X.}$  That is, the final topology ${\displaystyle \tau }$  is equal to the intersection ${\textstyle \tau =\bigcap _{i\in I}\tau _{i}.}$ 

Given a topological space ${\displaystyle (X,\tau )}$  and a family ${\displaystyle {\mathcal {C}}=\{C_{i}:i\in I\}}$  of subsets of ${\displaystyle X}$  each having the subspace topology, the final topology ${\displaystyle \tau _{\mathcal {C}}}$  induced by all the inclusion maps of the ${\displaystyle C_{i}}$  into ${\displaystyle X}$  is finer than (or equal to) the original topology ${\displaystyle \tau }$  on ${\displaystyle X.}$  The space ${\displaystyle X}$  is called coherent with the family ${\displaystyle {\mathcal {C}}}$  of subspaces if the final topology ${\displaystyle \tau _{\mathcal {C}}}$  coincides with the original topology ${\displaystyle \tau .}$  In that case, a subset ${\displaystyle U\subseteq X}$  will be open in ${\displaystyle X}$  exactly when the intersection ${\displaystyle U\cap C_{i}}$  is open in ${\displaystyle C_{i}}$  for each ${\displaystyle i\in I.}$  (See the coherent topology article for more details on this notion and more examples.) As a particular case, one of the notions of compactly generated space can be characterized as a certain coherent topology.

The direct limit of any direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms. Explicitly, this means that if ${\displaystyle \operatorname {Sys} _{Y}=\left(Y_{i},f_{ji},I\right)}$  is a direct system in the category Top of topological spaces and if ${\displaystyle \left(X,\left(f_{i}\right)_{i\in I}\right)}$  is a direct limit of ${\displaystyle \operatorname {Sys} _{Y}}$  in the category Set of all sets, then by endowing ${\displaystyle X}$  with the final topology ${\displaystyle \tau _{\mathcal {F}}}$  induced by ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\},}$  ${\displaystyle \left(\left(X,\tau _{\mathcal {F}}\right),\left(f_{i}\right)_{i\in I}\right)}$  becomes the direct limit of ${\displaystyle \operatorname {Sys} _{Y}}$  in the category Top.

The étalé space of a sheaf is topologized by a final topology.

A first-countable Hausdorff space ${\displaystyle (X,\tau )}$  is locally path-connected if and only if ${\displaystyle \tau }$  is equal to the final topology on ${\displaystyle X}$  induced by the set ${\displaystyle C\left([0,1];X\right)}$  of all continuous maps ${\displaystyle [0,1]\to (X,\tau ),}$  where any such map is called a path in ${\displaystyle (X,\tau ).}$ 

If a Hausdorff locally convex topological vector space ${\displaystyle (X,\tau )}$  is a Fréchet-Urysohn space then ${\displaystyle \tau }$  is equal to the final topology on ${\displaystyle X}$  induced by the set ${\displaystyle \operatorname {Arc} \left([0,1];X\right)}$  of all arcs in ${\displaystyle (X,\tau ),}$  which by definition are continuous paths ${\displaystyle [0,1]\to (X,\tau )}$  that are also topological embeddings.

Given functions ${\displaystyle f_{i}:Y_{i}\to X,}$  from topological spaces ${\displaystyle Y_{i}}$  to the set ${\displaystyle X}$ , the final topology on ${\displaystyle X}$  with respect to these functions ${\displaystyle f_{i}}$  satisfies the following property:

a function ${\displaystyle g}$  from ${\displaystyle X}$  to some space ${\displaystyle Z}$  is continuous if and only if ${\displaystyle g\circ f_{i}}$  is continuous for each ${\displaystyle i\in I.}$ 

 

This property characterizes the final topology in the sense that if a topology on ${\displaystyle X}$  satisfies the property above for all spaces ${\displaystyle Z}$  and all functions ${\displaystyle g:X\to Z}$ , then the topology on ${\displaystyle X}$  is the final topology with respect to the ${\displaystyle f_{i}.}$ 

Suppose ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:Y_{i}\to X\mid i\in I\right\}}$  is a family of maps, and for every ${\displaystyle i\in I,}$  the topology ${\displaystyle \upsilon _{i}}$  on ${\displaystyle Y_{i}}$  is the final topology induced by some family ${\displaystyle {\mathcal {G}}_{i}}$  of maps valued in ${\displaystyle Y_{i}}$ . Then the final topology on ${\displaystyle X}$  induced by ${\displaystyle {\mathcal {F}}}$  is equal to the final topology on ${\displaystyle X}$  induced by the maps ${\displaystyle \left\{f_{i}\circ g~:~i\in I{\text{ and }}g\in {\cal {G_{i}}}\right\}.}$ 

As a consequence: if ${\displaystyle \tau _{\mathcal {F}}}$  is the final topology on ${\displaystyle X}$  induced by the family ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\}}$  and if ${\displaystyle \pi :X\to (S,\sigma )}$  is any surjective map valued in some topological space ${\displaystyle (S,\sigma ),}$  then ${\displaystyle \pi :\left(X,\tau _{\mathcal {F}}\right)\to (S,\sigma )}$  is a quotient map if and only if ${\displaystyle (S,\sigma )}$  has the final topology induced by the maps ${\displaystyle \left\{\pi \circ f_{i}~:~i\in I\right\}.}$ 

By the universal property of the disjoint union topology we know that given any family of continuous maps ${\displaystyle f_{i}:Y_{i}\to X,}$  there is a unique continuous map $${\displaystyle f:\coprod _{i}Y_{i}\to X}$$  that is compatible with the natural injections. If the family of maps ${\displaystyle f_{i}}$  covers ${\displaystyle X}$  (i.e. each ${\displaystyle x\in X}$  lies in the image of some ${\displaystyle f_{i}}$ ) then the map ${\displaystyle f}$  will be a quotient map if and only if ${\displaystyle X}$  has the final topology induced by the maps ${\displaystyle f_{i}.}$ 

Throughout, let ${\displaystyle {\mathcal {F}}:=\left\{f_{i}:i\in I\right\}}$  be a family of ${\displaystyle X}$ -valued maps with each map being of the form ${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to X}$  and let ${\displaystyle \tau _{\mathcal {F}}}$  denote the final topology on ${\displaystyle X}$  induced by ${\displaystyle {\mathcal {F}}.}$  The definition of the final topology guarantees that for every index ${\displaystyle i,}$  the map ${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$  is continuous.

For any subset ${\displaystyle {\mathcal {S}}\subseteq {\mathcal {F}},}$  the final topology ${\displaystyle \tau _{\mathcal {S}}}$  on ${\displaystyle X}$  will be finer than (and possibly equal to) the topology ${\displaystyle \tau _{\mathcal {F}}}$ ; that is, ${\displaystyle {\mathcal {S}}\subseteq {\mathcal {F}}}$  implies ${\displaystyle \tau _{\mathcal {F}}\subseteq \tau _{\mathcal {S}},}$  where set equality might hold even if ${\displaystyle {\mathcal {S}}}$  is a proper subset of ${\displaystyle {\mathcal {F}}.}$ 

If ${\displaystyle \tau }$  is any topology on ${\displaystyle X}$  such that ${\displaystyle \tau \neq \tau _{\mathcal {F}}}$  and ${\displaystyle f_{i}:\left(Y_{i},\upsilon _{i}\right)\to (X,\tau )}$  is continuous for every index ${\displaystyle i\in I,}$  then ${\displaystyle \tau }$  must be strictly coarser than ${\displaystyle \tau _{\mathcal {F}}}$  (meaning that ${\displaystyle \tau \subseteq \tau _{\mathcal {F}}}$  and ${\displaystyle \tau \neq \tau _{\mathcal {F}};}$  this will be written ${\displaystyle \tau \subsetneq \tau _{\mathcal {F}}}$ ) and moreover, for any subset ${\displaystyle {\mathcal {S}}\subseteq {\mathcal {F}}}$  the topology ${\displaystyle \tau }$  will also be strictly coarser than the final topology ${\displaystyle \tau _{\mathcal {S}}}$  that ${\displaystyle {\mathcal {S}}}$  induces on ${\displaystyle X}$  (because ${\displaystyle \tau _{\mathcal {F}}\subseteq \tau _{\mathcal {S}}}$ ); that is, ${\displaystyle \tau \subsetneq \tau _{\mathcal {S}}.}$ 

Suppose that in addition, ${\displaystyle {\mathcal {G}}:=\left\{g_{a}:a\in A\right\}}$  is an ${\displaystyle A}$ -indexed family of ${\displaystyle X}$ -valued maps ${\displaystyle g_{a}:Z_{a}\to X}$  whose domains are topological spaces ${\displaystyle \left(Z_{a},\zeta _{a}\right).}$  If every ${\displaystyle g_{a}:\left(Z_{a},\zeta _{a}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$  is continuous then adding these maps to the family ${\displaystyle {\mathcal {F}}}$  will not change the final topology on ${\displaystyle X;}$  that is, ${\displaystyle \tau _{{\mathcal {F}}\cup {\mathcal {G}}}=\tau _{\mathcal {F}}.}$  Explicitly, this means that the final topology on ${\displaystyle X}$  induced by the "extended family" ${\displaystyle {\mathcal {F}}\cup {\mathcal {G}}}$  is equal to the final topology ${\displaystyle \tau _{\mathcal {F}}}$  induced by the original family ${\displaystyle {\mathcal {F}}=\left\{f_{i}:i\in I\right\}.}$  However, had there instead existed even just one map ${\displaystyle g_{a_{0}}}$  such that ${\displaystyle g_{a_{0}}:\left(Z_{a_{0}},\zeta _{a_{0}}\right)\to \left(X,\tau _{\mathcal {F}}\right)}$  was not continuous, then the final topology ${\displaystyle \tau _{{\mathcal {F}}\cup {\mathcal {G}}}}$  on ${\displaystyle X}$  induced by the "extended family" ${\displaystyle {\mathcal {F}}\cup {\mathcal {G}}}$  would necessarily be strictly coarser than the final topology ${\displaystyle \tau _{\mathcal {F}}}$  induced by ${\displaystyle {\mathcal {F}};}$  that is, ${\displaystyle \tau _{{\mathcal {F}}\cup {\mathcal {G}}}\subsetneq \tau _{\mathcal {F}}}$  (see this footnote^{[note 1]} for an explanation).

Let $${\displaystyle \mathbb {R} ^{\infty }~:=~\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to }}0\right\},}$$  denote the space of finite sequences, where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$  denotes the space of all real sequences. For every natural number ${\displaystyle n\in \mathbb {N} ,}$  let ${\displaystyle \mathbb {R} ^{n}}$  denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }}$  denote the inclusion map defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{n}\right):=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)}$  so that its image is $${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)=\left\{\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)~:~x_{1},\ldots ,x_{n}\in \mathbb {R} \right\}=\mathbb {R} ^{n}\times \left\{(0,0,\ldots )\right\}}$$  and consequently, $${\displaystyle \mathbb {R} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$$ 

Endow the set ${\displaystyle \mathbb {R} ^{\infty }}$  with the final topology ${\displaystyle \tau ^{\infty }}$  induced by the family ${\displaystyle {\mathcal {F}}:=\left\{\;\operatorname {In} _{\mathbb {R} ^{n}}~:~n\in \mathbb {N} \;\right\}}$  of all inclusion maps. With this topology, ${\displaystyle \mathbb {R} ^{\infty }}$  becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space. The topology ${\displaystyle \tau ^{\infty }}$  is strictly finer than the subspace topology induced on ${\displaystyle \mathbb {R} ^{\infty }}$  by ${\displaystyle \mathbb {R} ^{\mathbb {N} },}$  where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$  is endowed with its usual product topology. Endow the image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  with the final topology induced on it by the bijection ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right);}$  that is, it is endowed with the Euclidean topology transferred to it from ${\displaystyle \mathbb {R} ^{n}}$  via ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}.}$  This topology on ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  is equal to the subspace topology induced on it by ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right).}$  A subset ${\displaystyle S\subseteq \mathbb {R} ^{\infty }}$  is open (respectively, closed) in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$  if and only if for every ${\displaystyle n\in \mathbb {N} ,}$  the set ${\displaystyle S\cap \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  is an open (respectively, closed) subset of ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$  The topology ${\displaystyle \tau ^{\infty }}$  is coherent with the family of subspaces ${\displaystyle \mathbb {S} :=\left\{\;\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)~:~n\in \mathbb {N} \;\right\}.}$  This makes ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$  into an LB-space. Consequently, if ${\displaystyle v\in \mathbb {R} ^{\infty }}$  and ${\displaystyle v_{\bullet }}$  is a sequence in ${\displaystyle \mathbb {R} ^{\infty }}$  then ${\displaystyle v_{\bullet }\to v}$  in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$  if and only if there exists some ${\displaystyle n\in \mathbb {N} }$  such that both ${\displaystyle v}$  and ${\displaystyle v_{\bullet }}$  are contained in ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  and ${\displaystyle v_{\bullet }\to v}$  in ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$ 

Often, for every ${\displaystyle n\in \mathbb {N} ,}$  the inclusion map ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}}$  is used to identify ${\displaystyle \mathbb {R} ^{n}}$  with its image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  in ${\displaystyle \mathbb {R} ^{\infty };}$  explicitly, the elements ${\displaystyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}}$  and ${\displaystyle \left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)}$  are identified together. Under this identification, ${\displaystyle \left(\left(\mathbb {R} ^{\infty },\tau ^{\infty }\right),\left(\operatorname {In} _{\mathbb {R} ^{n}}\right)_{n\in \mathbb {N} }\right)}$  becomes a direct limit of the direct system ${\displaystyle \left(\left(\mathbb {R} ^{n}\right)_{n\in \mathbb {N} },\left(\operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\right)_{m\leq n{\text{ in }}\mathbb {N} },\mathbb {N} \right),}$  where for every ${\displaystyle m\leq n,}$  the map ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}$  is the inclusion map defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{m}\right):=\left(x_{1},\ldots ,x_{m},0,\ldots ,0\right),}$  where there are ${\displaystyle n-m}$  trailing zeros.

In the language of category theory, the final topology construction can be described as follows. Let ${\displaystyle Y}$  be a functor from a discrete category ${\displaystyle J}$  to the category of topological spaces Top that selects the spaces ${\displaystyle Y_{i}}$  for ${\displaystyle i\in J.}$  Let ${\displaystyle \Delta }$  be the diagonal functor from Top to the functor category Top^J (this functor sends each space ${\displaystyle X}$  to the constant functor to ${\displaystyle X}$ ). The comma category ${\displaystyle (Y\,\downarrow \,\Delta )}$  is then the category of co-cones from ${\displaystyle Y,}$  i.e. objects in ${\displaystyle (Y\,\downarrow \,\Delta )}$  are pairs ${\displaystyle (X,f)}$  where ${\displaystyle f=(f_{i}:Y_{i}\to X)_{i\in J}}$  is a family of continuous maps to ${\displaystyle X.}$  If ${\displaystyle U}$  is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set^J then the comma category ${\displaystyle \left(UY\,\downarrow \,\Delta ^{\prime }\right)}$  is the category of all co-cones from ${\displaystyle UY.}$  The final topology construction can then be described as a functor from ${\displaystyle \left(UY\,\downarrow \,\Delta ^{\prime }\right)}$  to ${\displaystyle (Y\,\downarrow \,\Delta ).}$  This functor is left adjoint to the corresponding forgetful functor.

• Direct limit – Special case of colimit in category theory
• Induced topology – Inherited topologyPages displaying short descriptions of redirect targets
• Initial topology – Coarsest topology making certain functions continuous
• LB-space
• LF-space – Topological vector space

• Brown, Ronald (June 2006). Topology and Groupoids. North Charleston: CreateSpace. ISBN 1-4196-2722-8.
• Willard, Stephen (1970). General Topology. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley. ISBN 9780201087079. Zbl 0205.26601.. (Provides a short, general introduction in section 9 and Exercise 9H)
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.