final topology

Given a set $X$ and an $I$-indexed family of topological spaces $\backslash left(Y\_i,\; \backslash upsilon\_i\backslash right)$ with associated functions

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

the {{em|final topology on $X$ induced by the family of functions $\backslash mathcal\{F\}\; :=\; \backslash left\backslash \{\; f\_i\; :\; i\; \backslash in\; I\; \backslash right\backslash \}$}} is the finest topology $\backslash tau\_\{\backslash mathcal\{F\}\}$ on $X$ such that

$$f\_i\; :\; \backslash left(Y\_i,\; \backslash upsilon\_i\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\}\backslash right)$$

is continuous for each $i\backslash in\; I$. The final topology always exists, and is unique.

Explicitly, the final topology may be described as follows:

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

The closed subsets have an analogous characterization:

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

The family $\backslash mathcal\{F\}$ of functions that induces the final topology on $X$ is usually a set of functions. But the same construction can be performed if $\backslash mathcal\{F\}$ 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 $\backslash mathcal\{G\}$ of $\backslash mathcal\{F\}$ with $\backslash mathcal\{G\}$ a set, such that the final topologies on $X$ induced by $\backslash mathcal\{F\}$ and by $\backslash mathcal\{G\}$ coincide. For more on this, see for example the discussion here.{{cite web |title=Set theoretic issues in the definition of k-space or final topology wrt a proper class of functions |url=https://math.stackexchange.com/questions/4639381 |website=Mathematics Stack Exchange}} 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.{{sfn|Brown|2006|loc=Section 5.9, p. 182}}

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

The final topology on a set $X$ induced by a family of $X$-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 $X\_i$, the disjoint union topology on the disjoint union $\backslash coprod\_i\; X\_i$ is the final topology on the disjoint union induced by the natural injections.

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

Given a topological space $(X,\backslash tau)$ and a family $\backslash mathcal\; C=\backslash \{\; C\_i\; :\; i\backslash in\; I\backslash \}$ of subsets of $X$ each having the subspace topology, the final topology $\backslash tau\_\{\backslash mathcal\; C\}$ induced by all the inclusion maps of the $C\_i$ into $X$ is finer than (or equal to) the original topology $\backslash tau$ on $X.$ The space $X$ is called coherent with the family $\backslash mathcal\; C$ of subspaces if the final topology $\backslash tau\_\{\backslash mathcal\; C\}$ coincides with the original topology $\backslash tau.$ In that case, a subset $U\backslash subseteq\; X$ will be open in $X$ exactly when the intersection $U\backslash cap\; C\_i$ is open in $C\_i$ for each $i\backslash in\; I.$ (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 $\backslash operatorname\{Sys\}\_Y\; =\; \backslash left(Y\_i,\; f\_\{ji\},\; I\backslash right)$ is a direct system in the category Top of topological spaces and if $\backslash left(X,\; \backslash left(f\_i\backslash right)\_\{i\; \backslash in\; I\}\backslash right)$ is a direct limit of $\backslash operatorname\{Sys\}\_Y$ in the category Set of all sets, then by endowing $X$ with the final topology $\backslash tau\_\{\backslash mathcal\{F\}\}$ induced by $\backslash mathcal\{F\}\; :=\; \backslash left\backslash \{\; f\_i\; :\; i\; \backslash in\; I\; \backslash right\backslash \},$ $\backslash left(\backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\}\backslash right),\; \backslash left(f\_i\backslash right)\_\{i\; \backslash in\; I\}\backslash right)$ becomes the direct limit of $\backslash operatorname\{Sys\}\_Y$ in the category Top.

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

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

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

Given functions $f\_i\; :\; Y\_i\; \backslash to\; X,$ from topological spaces $Y\_i$ to the set $X$, the final topology on $X$ with respect to these functions $f\_i$ satisfies the following property:

:a function $g$ from $X$ to some space $Z$ is continuous if and only if $g\; \backslash circ\; f\_i$ is continuous for each $i\; \backslash in\; I.$

Image:FinalTopology-01.png

This property characterizes the final topology in the sense that if a topology on $X$ satisfies the property above for all spaces $Z$ and all functions $g:X\backslash to\; Z$, then the topology on $X$ is the final topology with respect to the $f\_i.$

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

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

By the universal property of the disjoint union topology we know that given any family of continuous maps $f\_i\; :\; Y\_i\; \backslash to\; X,$ there is a unique continuous map

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

that is compatible with the natural injections.

If the family of maps $f\_i$ {{em|covers}} $X$ (i.e. each $x\; \backslash in\; X$ lies in the image of some $f\_i$) then the map $f$ will be a quotient map if and only if $X$ has the final topology induced by the maps $f\_i.$

Throughout, let $\backslash mathcal\{F\}\; :=\; \backslash left\backslash \{\; f\_i\; :\; i\; \backslash in\; I\; \backslash right\backslash \}$ be a family of $X$-valued maps with each map being of the form $f\_i\; :\; \backslash left(Y\_i,\; \backslash upsilon\_i\backslash right)\; \backslash to\; X$ and let $\backslash tau\_\{\backslash mathcal\{F\}\}$ denote the final topology on $X$ induced by $\backslash mathcal\{F\}.$

The definition of the final topology guarantees that for every index $i,$ the map $f\_i\; :\; \backslash left(Y\_i,\; \backslash upsilon\_i\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\}\backslash right)$ is continuous.

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

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

Suppose that in addition, $\backslash mathcal\{G\}\; :=\; \backslash left\backslash \{g\_a\; :\; a\; \backslash in\; A\backslash right\backslash \}$ is an $A$-indexed family of $X$-valued maps $g\_a\; :\; Z\_a\; \backslash to\; X$ whose domains are topological spaces $\backslash left(Z\_a,\; \backslash zeta\_a\backslash right).$

If every $g\_a\; :\; \backslash left(Z\_a,\; \backslash zeta\_a\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\}\backslash right)$ is continuous then adding these maps to the family $\backslash mathcal\{F\}$ will {{em|not}} change the final topology on $X;$ that is, $\backslash tau\_\{\backslash mathcal\{F\}\; \backslash cup\; \backslash mathcal\{G\}\}\; =\; \backslash tau\_\{\backslash mathcal\{F\}\}.$

Explicitly, this means that the final topology on $X$ induced by the "extended family" $\backslash mathcal\{F\}\; \backslash cup\; \backslash mathcal\{G\}$ is equal to the final topology $\backslash tau\_\{\backslash mathcal\{F\}\}$ induced by the original family $\backslash mathcal\{F\}\; =\; \backslash left\backslash \{\; f\_i\; :\; i\; \backslash in\; I\; \backslash right\backslash \}.$

However, had there instead existed even just one map $g\_\{a\_0\}$ such that $g\_\{a\_0\}\; :\; \backslash left(Z\_\{a\_0\},\; \backslash zeta\_\{a\_0\}\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\}\backslash right)$ was {{em|not}} continuous, then the final topology $\backslash tau\_\{\backslash mathcal\{F\}\; \backslash cup\; \backslash mathcal\{G\}\}$ on $X$ induced by the "extended family" $\backslash mathcal\{F\}\; \backslash cup\; \backslash mathcal\{G\}$ would necessarily be Comparison of topologies than the final topology $\backslash tau\_\{\backslash mathcal\{F\}\}$ induced by $\backslash mathcal\{F\};$ that is, $\backslash tau\_\{\backslash mathcal\{F\}\; \backslash cup\; \backslash mathcal\{G\}\}\; \backslash subsetneq\; \backslash tau\_\{\backslash mathcal\{F\}\}$ (see this footnoteBy definition, the map $g\_\{a\_0\}\; :\; \backslash left(Z\_\{a\_0\},\; \backslash zeta\_\{a\_0\}\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\}\backslash right)$ not being continuous means that there exists at least one open set $U\; \backslash in\; \backslash tau\_\{\backslash mathcal\{F\}\}$ such that $g\_\{a\_0\}^\{-1\}(U)$ is not open in $\backslash left(Z\_\{a\_0\},\; \backslash zeta\_\{a\_0\}\backslash right).$ In contrast, by definition of the final topology $\backslash tau\_\{\backslash mathcal\{F\}\; \backslash cup\; \backslash \{g\_\{a\_0\}\backslash \}\},$ the map $g\_\{a\_0\}\; :\; \backslash left(Z\_\{a\_0\},\; \backslash zeta\_\{a\_0\}\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\; \backslash cup\; \backslash \{g\_\{a\_0\}\backslash \}\}\backslash right)$ {{em|must}} be continuous. So the reason why $\backslash tau\_\{\backslash mathcal\{F\}\; \backslash cup\; \backslash mathcal\{G\}\}$ must be strictly coarser, rather than strictly finer, than $\backslash tau\_\{\backslash mathcal\{F\}\}$ is because the failure of the map $g\_\{a\_0\}\; :\; \backslash left(Z\_\{a\_0\},\; \backslash zeta\_\{a\_0\}\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_\{\backslash mathcal\{F\}\}\backslash right)$ to be continuous necessitates that one or more open subsets of $\backslash tau\_\{\backslash mathcal\{F\}\}$ must be "removed" in order for $g\_\{a\_0\}$ to become continuous. Thus $\backslash tau\_\{\backslash mathcal\{F\}\; \backslash cup\; \backslash \{g\_\{a\_0\}\backslash \}\}$ is just $\backslash tau\_\{\backslash mathcal\{F\}\}$ but some open sets "removed" from $\backslash tau\_\{\backslash mathcal\{F\}\}.$ for an explanation).

Let

$$\backslash R^\{\backslash infty\}\; ~:=~\; \backslash left\backslash \{\; \backslash left(x\_1,\; x\_2,\; \backslash ldots\backslash right)\; \backslash in\; \backslash R^\{\backslash N\}\; ~:~\; \backslash text\{\; all\; but\; finitely\; many\; \}\; x\_i\; \backslash text\{\; are\; equal\; to\; \}\; 0\; \backslash right\backslash \},$$

denote the {{em|space of finite sequences}}, where $\backslash R^\{\backslash N\}$ denotes the space of all real sequences.

For every natural number $n\; \backslash in\; \backslash N,$ let $\backslash R^n$ denote the usual Euclidean space endowed with the Euclidean topology and let $\backslash operatorname\{In\}\_\{\backslash R^n\}\; :\; \backslash R^n\; \backslash to\; \backslash R^\{\backslash infty\}$ denote the inclusion map defined by $\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right)\; :=\; \backslash left(x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; \backslash ldots\backslash right)$ so that its image is

$$\backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right)$$

= \left\{ \left(x_1, \ldots, x_n, 0, 0, \ldots\right) ~:~ x_1, \ldots, x_n \in \R \right\}

= \R^n \times \left\{ (0, 0, \ldots) \right\}

and consequently,

$$\backslash R^\{\backslash infty\}\; =\; \backslash bigcup\_\{n\; \backslash in\; \backslash N\}\; \backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right).$$

Endow the set $\backslash R^\{\backslash infty\}$ with the final topology $\backslash tau^\{\backslash infty\}$ induced by the family $\backslash mathcal\{F\}\; :=\; \backslash left\backslash \{\; \backslash ;\; \backslash operatorname\{In\}\_\{\backslash R^n\}\; ~:~\; n\; \backslash in\; \backslash N\; \backslash ;\; \backslash right\backslash \}$ of all inclusion maps.

With this topology, $\backslash R^\{\backslash infty\}$ becomes a complete Hausdorff locally convex sequential topological vector space that is {{em|not}} a Fréchet–Urysohn space.

The topology $\backslash tau^\{\backslash infty\}$ is strictly finer than the subspace topology induced on $\backslash R^\{\backslash infty\}$ by $\backslash R^\{\backslash N\},$ where $\backslash R^\{\backslash N\}$ is endowed with its usual product topology.

Endow the image $\backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right)$ with the final topology induced on it by the bijection $\backslash operatorname\{In\}\_\{\backslash R^n\}\; :\; \backslash R^n\; \backslash to\; \backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right);$ that is, it is endowed with the Euclidean topology transferred to it from $\backslash R^n$ via $\backslash operatorname\{In\}\_\{\backslash R^n\}.$

This topology on $\backslash operatorname\{Im\}\; \backslash left(\; \backslash operatorname\{In\}\_\{\backslash R^n\}\; \backslash right)$ is equal to the subspace topology induced on it by $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau^\{\backslash infty\}\backslash right).$

A subset $S\; \backslash subseteq\; \backslash R^\{\backslash infty\}$ is open (respectively, closed) in $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau^\{\backslash infty\}\backslash right)$ if and only if for every $n\; \backslash in\; \backslash N,$ the set $S\; \backslash cap\; \backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right)$ is an open (respectively, closed) subset of $\backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right).$

The topology $\backslash tau^\{\backslash infty\}$ is coherent with the family of subspaces $\backslash mathbb\{S\}\; :=\; \backslash left\backslash \{\; \backslash ;\; \backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right)\; ~:~\; n\; \backslash in\; \backslash N\; \backslash ;\; \backslash right\backslash \}.$

This makes $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau^\{\backslash infty\}\backslash right)$ into an LB-space.

Consequently, if $v\; \backslash in\; \backslash R^\{\backslash infty\}$ and $v\_\{\backslash bull\}$ is a sequence in $\backslash R^\{\backslash infty\}$ then $v\_\{\backslash bull\}\; \backslash to\; v$ in $\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau^\{\backslash infty\}\backslash right)$ if and only if there exists some $n\; \backslash in\; \backslash N$ such that both $v$ and $v\_\{\backslash bull\}$ are contained in $\backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right)$ and $v\_\{\backslash bull\}\; \backslash to\; v$ in $\backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right).$

Often, for every $n\; \backslash in\; \backslash N,$ the inclusion map $\backslash operatorname\{In\}\_\{\backslash R^n\}$ is used to identify $\backslash R^n$ with its image $\backslash operatorname\{Im\}\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right)$ in $\backslash R^\{\backslash infty\};$ explicitly, the elements $\backslash left(\; x\_1,\; \backslash ldots,\; x\_n\; \backslash right)\; \backslash in\; \backslash R^n$ and $\backslash left(x\_1,\; \backslash ldots,\; x\_n,\; 0,\; 0,\; 0,\; \backslash ldots\backslash right)$ are identified together.

Under this identification, $\backslash left(\backslash left(\backslash R^\{\backslash infty\},\; \backslash tau^\{\backslash infty\}\backslash right),\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^n\}\backslash right)\_\{n\; \backslash in\; \backslash N\}\backslash right)$ becomes a direct limit of the direct system $\backslash left(\backslash left(\backslash R^n\backslash right)\_\{n\; \backslash in\; \backslash N\},\; \backslash left(\backslash operatorname\{In\}\_\{\backslash R^m\}^\{\backslash R^n\}\backslash right)\_\{m\; \backslash leq\; n\; \backslash text\{\; in\; \}\; \backslash N\},\; \backslash N\backslash right),$ where for every $m\; \backslash leq\; n,$ the map $\backslash operatorname\{In\}\_\{\backslash R^m\}^\{\backslash R^n\}\; :\; \backslash R^m\; \backslash to\; \backslash R^n$ is the inclusion map defined by $\backslash operatorname\{In\}\_\{\backslash R^m\}^\{\backslash R^n\}\backslash left(x\_1,\; \backslash ldots,\; x\_m\backslash right)\; :=\; \backslash left(x\_1,\; \backslash ldots,\; x\_m,\; 0,\; \backslash ldots,\; 0\backslash right),$ where there are $n\; -\; m$ trailing zeros.

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

• {{annotated link|Direct limit}}
• {{annotated link|Induced topology}}
• {{annotated link|Initial topology}}
• {{annotated link|LB-space}}
• {{annotated link|LF-space}}

{{reflist|group=note}}

{{reflist}}

• {{cite book |last1=Brown |first1=Ronald |title=Topology and Groupoids. |date=June 2006 |publisher=CreateSpace |location=North Charleston |isbn=1-4196-2722-8}}
• {{cite book | first=Stephen | last=Willard | title=General Topology | url=https://archive.org/details/generaltopology00will_0 | url-access=registration | year=1970 | publisher=Addison-Wesley | location=Reading, MA | zbl=0205.26601 | series=Addison-Wesley Series in Mathematics| isbn=9780201087079 }}. (Provides a short, general introduction in section 9 and Exercise 9H)
• {{Willard General Topology}}

{{Topology|expanded}}

Category:General topology