Direct Limit Topology

In general topology and related areas of mathematics, the final topology (or coinduced,
strong, colimit, or inductive topology) on a set $X,$ with respect to a family of functions from topological spaces into $X,$ is the finest topology on $X$ that makes all those functions continuous.

The quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the topology that every direct limit in the category of topological spaces is endowed with, and it is in the context of direct limits that the final topology often appears. A topology is coherent with some collection of subspaces if and only if it is the final topology induced by the natural inclusions.

The dual notion is the initial topology, which for a given family of functions from a set $X$ into topological spaces is the coarsest topology on $X$ that makes those functions continuous.

Definition

Given a set $X$ and an $I$-indexed family of topological spaces $\left(Y_i, \upsilon_i\right)$ with associated functions
$f_i : Y_i \to X,$
the is the finest topology $\tau_$ on $X$ such that
$f_i : \left(Y_i, \upsilon_i\right) \to \left(X, \tau_\right)$

is continuous for each $i\in I$.

Explicitly, the final topology may be described as follows:
:a subset $U$ of $X$ is open in the final topology $\left(X, \tau_\right)$ (that is, $U \in \tau_$) if and only if $f_i^(U)$ is open in $\left(Y_i, \upsilon_i\right)$ for each $i\in I$.

The closed subsets have an analogous characterization:
:a subset $C$ of $X$ is closed in the final topology $\left(X, \tau_\right)$ if and only if $f_i^(C)$ is closed in $\left(Y_i, \upsilon_i\right)$ for each $i\in I$.

Examples

The important special case where the family of maps $\mathcal$ consists of a single surjective map can be completely characterized using the notion of quotient maps. A surjective function $f : (Y, \upsilon) \to \left(X, \tau\right)$ between topological spaces is a quotient map if and only if the topology $\tau$ on $X$ coincides with the final topology $\tau_$ induced by the family $\mathcal=\$. 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 $\coprod_i X_i$ is the final topology on the disjoint union induced by the natural injections.

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

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 $\operatorname_Y = \left(Y_i, f_, I\right)$ is a direct system in the category Top of topological spaces and if $\left(X, \left(f_i\right)_\right)$ is a direct limit of $\operatorname_Y$ in the category Set of all sets, then by endowing $X$ with the final topology $\tau_$ induced by $\mathcal := \left\,$ $\left(\left(X, \tau_\right), \left(f_i\right)_\right)$ becomes the direct limit of $\operatorname_Y$ in the category Top.

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

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

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

Properties

Characterization via continuous maps

Given functions $f_i : Y_i \to X,$ from topological spaces $Y_i$ to the set $X$, the final topology on $X$ can be characterized by the following property:
:a function $g$ from $X$ to some space $Z$ is continuous if and only if $g \circ f_i$ is continuous for each $i \in I.$

Behavior under composition

Suppose $\mathcal := \left\$ is a family of maps, and for every $i \in I,$ the topology $\upsilon_i$ on $Y_i$ is the final topology induced by some family $\mathcal_i$ of maps valued in $Y_i$. Then the final topology on $X$ induced by $\mathcal$ is equal to the final topology on $X$ induced by the maps $\left\.$

As a consequence: if $\tau_$ is the final topology on $X$ induced by the family $\mathcal := \left\$ and if $\pi : X \to (S, \sigma)$ is any surjective map valued in some topological space $(S, \sigma),$ then $\pi : \left(X, \tau_\right) \to (S, \sigma)$ is a quotient map if and only if $(S, \sigma)$ has the final topology induced by the maps $\left\.$

By the universal property of the disjoint union topology we know that given any family of continuous maps $f_i : Y_i \to X,$ there is a unique continuous map
$f : \coprod_i Y_i \to X$
that is compatible with the natural injections.
If the family of maps $f_i$ $X$ (i.e. each $x \in X$ 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 $f_i.$

Effects of changing the family of maps

Throughout, let $\mathcal := \left\$ be a family of $X$-valued maps with each map being of the form $f_i : \left(Y_i, \upsilon_i\right) \to X$ and let $\tau_$ denote the final topology on $X$ induced by $\mathcal.$
The definition of the final topology guarantees that for every index $i,$ the map $f_i : \left(Y_i, \upsilon_i\right) \to \left(X, \tau_\right)$ is continuous.

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

If $\tau$ is any topology on $X$ such that $\tau \neq \tau_$ and $f_i : \left(Y_i, \upsilon_i\right) \to (X, \tau)$ is continuous for every index $i \in I,$ then $\tau$ must be than $\tau_$ (meaning that $\tau \subseteq \tau_$ and $\tau \neq \tau_;$ this will be written $\tau \subsetneq \tau_$) and moreover, for any subset $\mathcal \subseteq \mathcal$ the topology $\tau$ will also be than the final topology $\tau_$ that $\mathcal$ induces on $X$ (because $\tau_ \subseteq \tau_$); that is, $\tau \subsetneq \tau_.$

Suppose that in addition, $\mathcal := \left\$ is an $A$-indexed family of $X$-valued maps $g_a : Z_a \to X$ whose domains are topological spaces $\left(Z_a, \zeta_a\right).$
If every $g_a : \left(Z_a, \zeta_a\right) \to \left(X, \tau_\right)$ is continuous then adding these maps to the family $\mathcal$ will change the final topology on $X;$ that is, $\tau_ = \tau_.$
Explicitly, this means that the final topology on $X$ induced by the "extended family" $\mathcal \cup \mathcal$ is equal to the final topology $\tau_$ induced by the original family $\mathcal = \left\.$
However, had there instead existed even just one map $g_$ such that $g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right)$ was continuous, then the final topology $\tau_$ on $X$ induced by the "extended family" $\mathcal \cup \mathcal$ would necessarily be than the final topology $\tau_$ induced by $\mathcal;$ that is, $\tau_ \subsetneq \tau_$ (see this footnoteBy definition, the map $g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right)$ not being continuous means that there exists at least one open set $U \in \tau_$ such that $g_^(U)$ is not open in $\left(Z_, \zeta_\right).$ In contrast, by definition of the final topology $\tau_,$ the map $g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right)$ be continuous. So the reason why $\tau_$ must be strictly coarser, rather than strictly finer, than $\tau_$ is because the failure of the map $g_ : \left(Z_, \zeta_\right) \to \left(X, \tau_\right)$ to be continuous necessitates that one or more open subsets of $\tau_$ must be "removed" in order for $g_$ to become continuous. Thus $\tau_$ is just $\tau_$ but some open sets "removed" from $\tau_.$ for an explanation).

Coherence with subspaces

Let $(X, \tau)$ be a topological space and let $\mathbb$ be a family of subspaces of $(X, \tau)$ where importantly, the word "sub" is used to indicate that each subset $S \in \mathbb$ is endowed with the subspace topology $\tau\vert_S$ inherited from $(X, \tau).$
The space $(X, \tau)$ is said to be with the family $\mathbb$ of subspaces if $\tau = \tau_,$ where $\tau_$ denotes the final topology induced by the inclusion maps $\mathcal := \left\$ where for every $S \in \mathbb,$ the inclusion map takes the form
:$\operatorname_S^X : \left( S, \tau\vert_S \right) \to X.$

Unraveling the definition, $(X, \tau)$ is coherent with $\mathbb$ if and only if the following statement is true:
:for every subset $U \subseteq X,$ $U$ is open in $(X, \tau)$ if and only if for every $S \in \mathbb,$ $U \cap S$ is open in the subspace $\left(S, \tau\vert_S\right).$

Closed sets can be checked instead: $(X, \tau)$ is coherent with $\mathbb$ if and only if for every subset $C \subseteq X,$ $C$ is closed in $(X, \tau)$ if and only if for every $S \in \mathbb,$ $C \cap S$ is closed in $\left( S, \tau\vert_S \right).$

For example, if $\mathbb$ is a cover of a topological space $(X, \tau)$ by open subspaces (i.e. open subsets of $(X, \tau)$ endowed with the subspace topology) then $\tau$ is coherent with $\mathbb.$
In contrast, if $\mathbb$ is the set of all singleton subsets of $(X, \tau)$ (each set being endowed with its unique topology) then $(X, \tau)$ is coherent with $\mathbb$ if and only if $\tau$ is the discrete topology on $X.$
The disjoint union is the final topology with respect to the family of canonical injections.
A space $(X, \tau)$ is called and a if $\tau$ is coherent with the set $\mathbb$ of all compact subspaces of $(X, \tau).$
All first-countable spaces and all Hausdorff locally compact spaces are -spaces, so that in particular, every manifold and every metrizable space is coherent with the family of all its compact subspaces.

As demonstrated by the examples that follows, under certain circumstance, it may be possible to characterize a more general final topology in terms of coherence with subspaces.
Let $\mathcal := \left\$ be a family of $X$-valued maps with each map being of the form $f_i : \left(Y_i, \upsilon_i\right) \to X$ and let $\tau_$ denote the final topology on $X$ induced by $\mathcal.$
Suppose that $\tau$ is a topology on $X$ and for every index $i \in I,$ the image $\operatorname f_i := f_i(X)$ is endowed with the subspace topology $\tau\vert_$ inherited from $(X, \tau).$
If for every $i \in I,$ the map $f_i : \left(Y_i, \upsilon_i\right) \to \left( \operatorname f_i, \tau\vert_ \right)$ is a quotient map then $\tau = \tau_$ if and only if $(X, \tau)$ is coherent with the set of all images $\left\.$

Final topology on the direct limit of finite-dimensional Euclidean spaces

Let
$\R^ ~:=~ \left\,$
denote the , where $\R^$ denotes the space of all real sequences.
For every natural number $n \in \N,$ let $\R^n$ denote the usual Euclidean space endowed with the Euclidean topology and let $\operatorname_ : \R^n \to \R^$ denote the inclusion map defined by $\operatorname_\left(x_1, \ldots, x_n\right) := \left(x_1, \ldots, x_n, 0, 0, \ldots\right)$ so that its image is
$\operatorname \left(\operatorname_\right) = \left\ = \R^n \times \left\$
and consequently,
$\R^ = \bigcup_ \operatorname \left(\operatorname_\right).$

Endow the set $\R^$ with the final topology $\tau^$ induced by the family $\mathcal := \left\$ of all inclusion maps.
With this topology, $\R^$ becomes a complete Hausdorff locally convex sequential topological vector space that is a Fréchet–Urysohn space.
The topology $\tau^$ is strictly finer than the subspace topology induced on $\R^$ by $\R^,$ where $\R^$ is endowed with its usual product topology.
Endow the image $\operatorname \left(\operatorname_\right)$ with the final topology induced on it by the bijection $\operatorname_ : \R^n \to \operatorname \left(\operatorname_\right);$ that is, it is endowed with the Euclidean topology transferred to it from $\R^n$ via $\operatorname_.$
This topology on $\operatorname \left( \operatorname_ \right)$ is equal to the subspace topology induced on it by $\left(\R^, \tau^\right).$
A subset $S \subseteq \R^$ is open (respectively, closed) in $\left(\R^, \tau^\right)$ if and only if for every $n \in \N,$ the set $S \cap \operatorname \left(\operatorname_\right)$ is an open (respectively, closed) subset of $\operatorname \left(\operatorname_\right).$
The topology $\tau^$ is coherent with the family of subspaces $\mathbb := \left\.$
This makes $\left(\R^, \tau^\right)$ into an LB-space.
Consequently, if $v \in \R^$ and $v_$ is a sequence in $\R^$ then $v_ \to v$ in $\left(\R^, \tau^\right)$ if and only if there exists some $n \in \N$ such that both $v$ and $v_$ are contained in $\operatorname \left(\operatorname_\right)$ and $v_ \to v$ in $\operatorname \left(\operatorname_\right).$

Often, for every $n \in \N,$ the inclusion map $\operatorname_$ is used to identify $\R^n$ with its image $\operatorname \left(\operatorname_\right)$ in $\R^;$ explicitly, the elements $\left( x_1, \ldots, x_n \right) \in \R^n$ and $\left(x_1, \ldots, x_n, 0, 0, 0, \ldots\right)$ are identified together.
Under this identification, $\left(\left(\R^, \tau^\right), \left(\operatorname_\right)_\right)$ becomes a direct limit of the direct system $\left(\left(\R^n\right)_, \left(\operatorname_^\right)_, \N\right),$ where for every $m \leq n,$ the map $\operatorname_^ : \R^m \to \R^n$ is the inclusion map defined by $\operatorname_^\left(x_1, \ldots, x_m\right) := \left(x_1, \ldots, x_m, 0, \ldots, 0\right),$ where there are $n - m$ trailing zeros.

Categorical description

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 $i \in J.$ Let $\Delta$ 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 $(Y \,\downarrow\, \Delta)$ is then the category of co-cones from $Y,$ i.e. objects in $(Y \,\downarrow\, \Delta)$ are pairs $(X, f)$ where $f_i : Y_i \to X$ is a family of continuous maps to $X.$ If $Y$ is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set^''J'' then the comma category $\left(UY \,\downarrow\, \Delta^\right)$ is the category of all co-cones from $UY.$ The final topology construction can then be described as a functor from $\left(UY \,\downarrow\, \Delta^\right)$ to $(Y \,\downarrow\, \Delta).$ This functor is left adjoint to the corresponding forgetful functor.

See also

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Notes

Citations

References

* . ''(Provides a short, general introduction in section 9 and Exercise 9H)''
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{{Topology, expanded

General topology