Initial topology

 ( General Topology ) 

In general topology and related areas of mathematics, the initial topology (or induced topology

or strong topology or limit topology or projective topology) on a set $X,$ with respect to a family of functions on $X,$ is the coarsest topology on $X$ that makes those functions 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 dual notion is the final topology, which for a given family of functions mapping to a set $Y$ is the finest topology on $Y$ that makes those functions continuous.

---

Definition 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 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 subbase $\backslash bigcup\_\{i\; \backslash in\; I\}\; \backslash tau\_i.$

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 is the least upper bound topology of the $I$-indexed family of topologies $f\_i^\{-1\}\backslash left(\backslash sigma\_i\backslash right)$ (for $i\; \backslash in\; I$). Explicitly, the initial topology is the collection of open sets subbase 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 . If $I$ contains Singleton set, then all the open sets of the initial topology $(X,\; \backslash tau)$ 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 .
• 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 indexed 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 function) 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.

---

Properties

---

Characteristic property 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 $i\; \backslash in\; I.$

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 $i\; \backslash in\; I.$

---

Evaluation 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

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

\end{alignat}

This map is known as the .

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 $X.$

---

Hausdorffness 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 $X.$

---

Transitivity of the initial topology 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 $J\_i.$ 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 $S.$

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 $I.$ 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 $f\; :\; X\; \backslash to\; \backslash prod\_i\; Y\_i\backslash ,.$ 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 $\backslash prod\_i\; Y\_i.$

---

Separating points from closed sets 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 $A$ 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.

---

Initial uniform structure 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 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 $\{\backslash textstyle\; \backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\; \backslash mathcal\{U\}\_i\}.$ 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 $\backslash left(\backslash tau\_i\backslash right)\_\{i\; \backslash in\; I\}.$

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 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. 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 $i\; \backslash in\; I$). 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. 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.

Hausdorffness: The topology on $X$ induced by the initial uniform structure $\backslash mathcal\{U\}$ is Hausdorff space 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 $\backslash left(f\_i(x),\; f\_i(y)\backslash right)\; \backslash not\backslash in\; V\_i.$ 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 $X$ (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 $i\; \backslash in\; I.$

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 $i\; \backslash in\; I.$

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.

---

Categorical description 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: 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 adjoint functor 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 $\backslash mathrm\{Cone\}(UY).$

---

See also

---

Bibliography

• Stephen Willard (1970). 9780486434797, Addison-Wesley. . ISBN 9780486434797

---

External links

Post Comment