Codex @codex  0

For arbitrary topological spaces $X_i$, an index set $I$, a sequence $x^{(n)}\in {\prod }_i{X}_i$, and a point $x$ in that Cartesian product, convergence in a product space to $x$ in the product topology is equivalent to coordinatewise convergence in every factor. No countability or Hausdorff assumption on the factors is needed. For a product space of copies of the real line, regarding each point as a function $I\to \mathbb{R}$ identifies this with pointwise convergence.

Proof outline: the forward implication follows because each projection map is continuous. Conversely, given a basic neighborhood $B(F,U)$ of $x$, each of its finitely many coordinate tests holds for all $n\ge N_i$. Taking $N={\max }_{i\in F}{N}_i$ makes all tests hold simultaneously; if $F$ is empty, any threshold works. This proves eventual membership in every neighborhood, rather than only in a sampled collection of cylinders. The same equivalence holds for nets: a directed set supplies a common upper bound for finitely many thresholds. Sequences alone need not characterize the whole topology of an arbitrary product space.

A partially ordered set is directed-complete if every directed set in it has a least upper bound. A complete lattice is directed-complete. Conversely, directed completeness together with all finite joins, including the empty join, gives arbitrary joins: take the directed set of joins of finite subsets.

A direct system indexed by a directed set $A$ consists of abelian groups $G_a$ and homomorphisms ${\varphi }_{ab}:G_a\to G_b$ for $a\le b$, with ${\varphi }_{aa}=1$ and ${\varphi }_{bc}{\varphi }_{ab}={\varphi }_{ac}$.

Use the pointed complete partial order convention: a partially ordered set has a least element and every nonempty directed set has a least upper bound. The argument also works for the convention requiring only suprema of increasing countable chains. The function space of complete partial orders $[D\to E]$ consists of Scott continuous maps, namely monotone maps preserving directed suprema, ordered pointwise. Its least element is the constant map with value ${\perp }_E$.

For a directed family $\{f_i\}$, define $f(x)={\sup }_i{f}_i(x)$. This is its pointwise directed supremum and is monotone. If $A\subseteq D$ is directed, thenThe interchange is justified because $\{f_i(a):i,a\}$ is directed: choose a common upper map and a common upper input. Thus $f$ is Scott continuous, proving that $[D\to E]$ is a cpo. Evaluation is also jointly Scott continuous: for a directed family $(f_i,x_i)$, the pairs of independently chosen indices are cofinal in a common larger index, so $({\sup }_i{f}_i)({\sup }_i{x}_i)={\sup }_i{f}_i(x_i)$. The same pointwise calculation proves continuity of currying.

Suppose the continuous maps $e:[D\to D]\to D$, $p:D\to [D\to D]$ satisfy $p\circ e=\mathrm{id}$. DefineFor an environment $\rho$, interpret a variable by $\rho (x)$, application by $\cdot$, and an abstraction byInduction on syntax, using evaluation, currying and continuity of $e,p$, shows that every environment-dependent interpretation is Scott continuous, so this abstraction is defined. A further syntax induction proves the substitution lemmaFor abstraction the bound variable is first renamed fresh. Consequently $p(e(f))=f$ proves the beta equation for every environment. Consistent renaming proves alpha invariance, and the compositional interpretation respects all contexts. This is a lambda model. A retraction alone does not assert $e\circ p=\mathrm{id}$, so eta conversion need not be valid.

For the graph model of lambda calculus, take $P\omega =\mathcal{P}(\mathbb{N})$ with inclusion, bottom $\varnothing$ and directed supremum equal to union. Let $e_k$ be the finite set of positions of one-bits in $k$, and choose the pairing functionDefineA finite witness $e_k$ lies in one member of any directed union containing it, proving continuity in $b$; a graph entry lies in one member of any directed union of graphs, proving continuity in $a$. The displayed definition of $e$ preserves pointwise directed unions as well. Finallysince the finite subsets of $b$ are directed with union $b$. This proves the retraction and all claimed model properties. The composite $e\circ p$ generally enlarges a graph by adding entries with larger finite-input witnesses; it is not generally the identity.

A preframe has finite meets and joins of nonempty directed sets, with finite meets distributing over these joins. A preframe homomorphism preserves both operations. Its preframe tensor product $P\#Q$ has generators $p\#q$ and relations making $(p,q)\mapsto p\#q$ a homomorphism separately in each variable. Equivalently,In particular $1\#q=p\#1=1$. The tensor unit is the free preframe on one generator: classically it is $2$, and constructively it is the frame $\Omega$ of truth values, with generator $0$.

Regard a frame $L$ as a commutative monoid object in preframes, with multiplication $x\ast y=x\vee y$ and unit $0$. Binary join preserves finite meets and directed joins separately. Conversely take a commutative monoid object $(M,\ast ,e)$ in preframes and putThis set contains $e$ and $1$, since preservation of the empty meet gives $1\ast x=1$. If $x,y\in R(M)$, then $x\ast y\ge x,y$, andThus finite meets are inherited. If $D\subseteq R(M)$ is nonempty and directed, separate preservation of directed joins givesbecause a common upper bound $t\in D$ gives $d\ast d^{\prime }\le t\ast t=t$. Directed joins are inherited too. Associativity and commutativity show that $x\ast y$ is idempotent; it is the least idempotent upper bound of $x,y$. Therefore $e$ is the bottom and $\ast$ is binary join on $R(M)$. Arbitrary joins are directed joins of finite products, including the empty product $e$. Finallyso binary distributivity, together with directed distributivity, proves that $R(M)$ is a frame.

A homomorphism from a frame join monoid to $M$ sends every element to an idempotent above $e$ and therefore factors uniquely through $R(M)$. The factor preserves finite joins and directed joins, hence arbitrary joins. Conversely the inclusion $R(M)\to M$ preserves the preframe and monoid operations. Also a preframe monoid homomorphism between frames is automatically a frame homomorphism. This proves the full embedding and its right adjoint: $\mathbf{Frm}$ is a coreflective subcategory of $\mathbf{CMon}(\mathbf{PFrm})$, with coreflector $R$.

The coproduct of commutative monoid objects in a symmetric monoidal category is their tensor: the maps are $x\mapsto x\#e_N$, $y\mapsto e_M\#y$, and a pair $f,g$ induces $x\#y\mapsto f(x)\ast g(y)$. For two frames $L,K$, the induced multiplication on pure generators isEvery pure generator is idempotent and above $e$. Those properties are preserved by finite meets and directed joins, as just proved, so hold throughout the generated preframe. It is therefore already a frame, rather than requiring a further coreflection. ConsequentlyRectangles in the localic product are instead $i_L(a)\wedge i_K(b)=(a\#0)\wedge (0\#b)$.

For a filtered colimit in a category of frames, form the colimit $P$ in preframes, with stage maps $i_j$. It can be presented by stage elements, all their finite-meet and directed-join relations, and the transition relations. Since tensoring is a left adjoint, it preserves colimits. Filteredness makes the diagonal diagram cofinal in the product index diagram, so the stage multiplications induce $P\#P\to P$. The units agree. Each stage element is idempotent above the common unit; generation again implies $P=R(P)$. The resulting frame is the filtered colimit in a category in $\mathbf{Frm}$; its binary join is induced by computing two representatives at a common later stage. Thus binary coproducts and filtered colimits of frames are computed in preframes, with the indicated induced join operation.

We now prove compactness. For a compact locale, its frame $L$ has a top test $q_L:L\to \Omega$, $q_L(a)=[a=1]$, preserving finite meets and directed joins. For compact $L,K$, tensor the tests and use the unit isomorphism $\Omega \#\Omega \cong \Omega$ to obtain a preframe map $q:L\#K\to \Omega$ withIt reflects top on every pure generator. The property $q(x)=1⟹x=1$ is closed under finite meets and directed joins: for a directed join, preservation of the join says that its truth value is the existential disjunction of the individual truth values, so some term has value $1$. Since pure generators generate the preframe, $q$ reflects top everywhere. Applying $q$ to a directed cover of $1$ therefore shows that a term of that cover is $1$. This proves compactness of the binary coproduct frame, hence of the binary product locale.

For a filtered diagram of compact frames, define on stage $j$These truth-valued maps preserve finite meets: witnesses can be taken to a common later stage, using filteredness. They preserve directed joins: compactness at the witnessing later stage selects a member of the directed family. They are compatible with transitions, again by filteredness, so induce a preframe map $q:P\to \Omega$. On stage generators, $q(i_j(a))=1$ implies $i_j(a)=1$. The same generation argument proves that $q$ reflects top throughout $P$. Thus the filtered colimit frame is compact.

Finally the arbitrary coproduct of the frames $\mathcal{O}(X_i)$ is the filtered colimit of their finite coproducts, indexed by finite subsets of the index set; the empty finite coproduct is $\Omega$, itself compact. Finite coproducts are compact by the binary argument, and their filtered colimit is compact by the preceding paragraph. Opposing the categories givesThis proves the localic Tychonoff theorem through preframes, without choosing points or invoking the axiom of choice.

An inverse system of topological spaces over a directed set $I$ consists of spaces $X_i$ and continuous maps $f_{ij}:X_j\to X_i$ for $i\le j$, with $f_{ii}=1$ and $f_{ik}=f_{ij}{f}_{jk}$. Its inverse limit is a space $X$ with compatible continuous projections ${\pi }_i$ such that every compatible family of continuous maps $u_i:Y\to X_i$ factors uniquely through a continuous map $u:Y\to X$. To construct it, takewith the subspace topology inherited from the product topology. The map $u(y)=(u_i(y))$ has values in $X$ and is continuous by the definition of the product topology; its coordinates force uniqueness. If $X^{\prime }$ is another inverse limit, the two universal properties give continuous maps $X\to X^{\prime }$ and $X^{\prime }\to X$ whose composites have all the same projections as the identity. Uniqueness makes these composites the identity. The inverse limit exists and is unique up to a unique projection-preserving homeomorphism. An arbitrary inverse limit may be empty; nonemptiness requires additional hypotheses.

Use the usual Hausdorff convention for a topological group. An inverse limit of finite discrete groups is a closed subgroup of their product: each compatibility condition is closed because the finite factors are Hausdorff spaces. The Tychonoff theorem gives compactness, and distinct tuples are separated by a clopen coordinate cylinder, so the limit is totally disconnected. Coordinatewise multiplication and inversion are continuous, giving a profinite group.

Conversely, let $G$ be a compact Hausdorff space and a totally disconnected topological group. We use the standard topological fact that a compact Hausdorff space that is totally disconnected has a basis of clopen sets. Given an identity neighbourhood, choose a clopen identity neighbourhood $U$ inside it. Compactness of $U$ and continuity of multiplication give a symmetric identity neighbourhood $V$ with $VU\subseteq U$: cover $U$ by finitely many neighbourhoods on which multiplication by a sufficiently small identity neighbourhood stays inside $U$, then intersect those identity neighbourhoods. Symmetry gives $vU=U$ for $v\in V$. Consequently $H=\{g:gU=U\}$ is a subgroup containing $V$, hence an open subgroup, and $H\subseteq U$ since $1\in U$. Its open cosets cover the compact group, so there are finitely many. The normal core $N={\bigcap }_{g\in G}gH{g}^{-1}$ is a finite intersection of open subgroups, hence an open normal subgroup contained in $U$. Thus the open normal subgroups form an identity-neighbourhood basis and have trivial intersection.

The canonical group homomorphism $G\to {\underleftarrow{\lim }}_{N}G/N$ is continuous and injective. For a compatible tuple of quotient elements, its inverse-image cosets in $G$ are closed and have the finite intersection property: a finite family is refined by the quotient for the intersection of its normal subgroups. Compactness gives an element in all those cosets, proving surjectivity. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. The compact totally disconnected and inverse-limit definitions of a profinite group agree.

Index the finite nonempty sets $\mathcal{P}(N)$ by open normal subgroups, with the transition from $M$ to $N$ when $M\subseteq N$ given by image under $G/M\to G/N$. Images of Sylow subgroups under surjections of finite groups are Sylow subgroups. These transitions are surjective: if $S\le G/N$ is a Sylow subgroup, choose a Sylow subgroup in its full inverse image in $G/M$. It maps onto $S$, and its order has the full $p$-part of $|G/M|$, so it is a Sylow subgroup of $G/M$ too. The transitions respect composition, giving an inverse system of finite sets.

Its inverse limit is nonempty. Indeed, finitely many compatibility conditions can be satisfied by choosing one Sylow subgroup in a quotient refining all the involved quotients and taking its images; compactness of the product of the finite sets then gives a compatible family $(P_N{)}_N$. DefineThis is a closed subgroup of $G$. Its image in $G/N$ is exactly $P_N$: for a prescribed $s\in P_N$, each finite collection of conditions defining $P$ together with ${\pi }_N(g)=s$ is satisfiable in a common refining quotient, because the transitions on the chosen Sylow subgroups are surjective. The finite intersection property again supplies $g\in P$. Hence $P\cong {\underleftarrow{\lim }}_N{P}_N$ is a pro-p group and $PN/N=P_N$ for every $N$.

Finally, if a pro-p subgroup $Q$ contains $P$, its image in every finite quotient is a p-group containing the Sylow subgroup $P_N$, so that image equals $P_N$. Thus every $q\in Q$ belongs to every ${\pi }_N^{-1}(P_N)$, whence $Q\subseteq P$. The constructed $P$ is a maximal pro-p subgroup.

A topological group is a group with a topology for which multiplication $G\times G\to G$ and inversion $G\to G$ are continuous. A profinite group has either of the following equivalent descriptions: a compact Hausdorff totally disconnected topological group, or an inverse limit of finite discrete groups. Totally disconnected means that every connected component is a singleton.

For clarity, an inverse system over a directed set $I$ consists of groups $G_i$ and maps ${\varphi }_{ij}:G_j\to G_i$ for $i\le j$, with ${\varphi }_{ii}=1$ and ${\varphi }_{ik}={\varphi }_{ij}{\varphi }_{jk}$. Its inverse limit iswith coordinatewise multiplication and the subspace topology. Equivalently, it is a group with compatible continuous projections such that every other compatible family of maps into the $G_i$ factors uniquely through it.

The displayed limit is a closed subgroup of the compact Hausdorff product of the finite groups, and is totally disconnected, since two different tuples are separated by a clopen coordinate condition. Conversely a compact Hausdorff totally disconnected group has a neighbourhood basis of open normal subgroups. One way to see the group part of this standard clopen-basis fact is to choose a clopen identity neighbourhood $U$ inside a prescribed neighbourhood. Compactness of $U$ and continuity give a symmetric identity neighbourhood $V$ with $VU\subseteq U$. The stabilizer $H=\{g:gU=U\}$ contains $V$, is open, and is contained in $U$ because $1\in U$. Its index is finite by part (ii); its normal core is the intersection of finitely many conjugates and is open and normal. The canonical mapis injective because these $N$ separate points, and surjective because compatible cosets have the finite intersection property and $G$ is compact. It is a homeomorphism by compactness and the Hausdorff property. This explains the equivalence of the two definitions.

The profinite Frattini subgroup is $\Phi (G)=\bigcap M$, where $M$ ranges over all maximal proper open subgroups; for the trivial group the empty intersection is $G$. For a pro-p group,Here powers and group commutators mean generated subgroups and the bar supplies closure. Every maximal open subgroup in a pro-p group is normal of index $p$, as follows by passing to the finite p-group quotient by its normal core. Such a subgroup therefore contains all powers and group commutators. Conversely the quotient by $D=\overline{G^p[G,G]}$ is an abelian pro-p group of exponent $p$. If an element of this quotient is nontrivial, some finite elementary abelian quotient detects it; a linear functional to ${\mathbb{F}}_p$ then detects it as well. Its group homomorphism kernel pulls back to a maximal open subgroup of $G$. Thus the intersection of these kernels is exactly $D$.

We also use the topological generation criterion: a subset generates $G$ topologically if and only if its image generates $G/\Phi (G)$ topologically. To prove the nontrivial direction, a proper closed generated subgroup $K$ lies in a proper open subgroup $KN$ for a suitably small open normal $N$, and hence in a maximal open subgroup obtained from the finite quotient $G/N$. Since that maximal subgroup contains $\Phi (G)$, the image of $K$ cannot generate the Frattini quotient.

Now suppose $G$ is a nontrivial procyclic pro-p group. If it had distinct maximal open subgroups $M_1,M_2$, the homomorphism to $C_p\times C_p$ induced by the two quotient maps would be onto: its projections are onto, and a proper such subgroup would be a one-dimensional graph, forcing the two kernels to coincide. Hence $G/(M_1\cap M_2)\cong C_p^2$, contradicting procyclicity. Thus there is exactly one maximal open subgroup $M$, and $\Phi (G)=M$. Any $g\notin M$ generates $G/\Phi (G)\cong C_p$, so the criterion givesFor the trivial group use its identity as generator. The additive group ${\mathbb{Z}}_p$ is an infinite example, topologically generated by $1$: its finite continuous quotients are cyclic $\mathbb{Z}/p^n\mathbb{Z}$.

Here a pointed complete partial order is a partially ordered set $D$ with a least element ${\perp }_D$ in which every nonempty directed set has a least upper bound. Directed means that every pair of its elements has an upper bound within that set. A Scott continuous map is monotone and preserves these directed suprema. It is not required to preserve the least element; restricting to strict maps would change the categorical claim. If completeness is instead formulated only for increasing countable chains, the same arguments below work with those chains.

Order the function space of complete partial orders $[D\to E]$ pointwise: $f\le g$ means $f(x)\le g(x)$ for every $x$. Its least element is the constant function with value ${\perp }_E$. For a directed family $\mathcal{F}$ of continuous maps, define $h(x)={\sup }_{f\in \mathcal{F}}f(x)$. This is well-defined and is its pointwise least upper bound. It is monotone. For any directed $A\subseteq D$,The interchange is valid because the combined family $\{f(a):f\in \mathcal{F},a\in A\}$ is directed, and either iterated supremum is its least upper bound. Thus $h$ is a Scott continuous map, proving that $[D\to E]$ is a pointed complete partial order.

Finite products carry the product order, componentwise suprema and least element $({\perp }_D,{\perp }_E)$. The one-element order is a terminal object. For the exponential, use the function space with evaluation $\mathrm{ev}(f,x)=f(x)$. To check joint continuity, take a directed family of pairs $(f_i,x_i)$. ThenIndeed, directedness supplies a $k$ above the pairs indexed by both $i$ and $j$, giving $f_i(x_j)\le f_k(x_k)$. Thus the diagonal values are cofinal in the double family, and evaluation preserves its supremum.

For a continuous $h:A\times D\to E$, its currying is $\Lambda h:A\to [D\to E]$, $(\Lambda h)(a)(d)=h(a,d)$. Each section is continuous because $d\mapsto (a,d)$ is; pointwise directed suprema in the $a$ coordinate show that $\Lambda h$ is continuous. Conversely a continuous $g:A\to [D\to E]$ gives the continuous map $\mathrm{ev}\circ (g\times {\mathrm{id}}_D)$. The two constructions are inverse, because they agree on every pair $(a,d)$. This gives the natural bijectionwith the required evaluation map. Together with the terminal object and products, it proves cartesian closure of pointed complete partial orders with non-strict continuous maps.

A complete lattice has every join, hence every directed and finite join. Conversely, if a partially ordered set is directed-complete and has finite joins including the empty join, then for any subset $A$ the set of joins of its finite subsets is nonempty and directed. Its least upper bound is $\bigvee A$. Including the empty join supplies a bottom element and handles $A=\varnothing$.

For a directed set of partial functions from $A$ to $B$, their union is a function: any two functions have a common extension, so they agree wherever their domains overlap. This union is exactly their least upper bound in the extension order, proving directed completeness.

Write $C_x$ for the intersection of all sets containing $x$ and closed under $f$ and directed joins. Then $xRy$ means $y\in C_x$. Reflexivity is immediate, and transitivity follows because any closed set containing $x$ must contain $y$ and then every $z$ with $yRz$. The upper set $[x,\infty )$ is closed: $f$ is inflationary, and a directed join of elements above $x$ is above $x$. Thus $xRy$ implies $x\le y$, proving antisymmetry.

If $h,k\in H$, then $xRk(x)Rh(k(x))$, so $h\circ k\in H$. Moreover $h(k(x))\ge h(x)$ by monotonicity, and $h(k(x))\ge k(x)$ by inflationarity. This proves that $Hx$ is directed. Define $h_0(x)=\bigvee Hx$. Pointwise monotonicity of every $h$ implies monotonicity of $h_0$. Since $Hx\subseteq C_x$ and $C_x$ is closed under directed joins, $xR{h}_0(x)$; hence $h_0\in H$.

The original map $f$ belongs to $H$ by the definition of closedness, so $f\circ h_0\in H$ and $f(h_0(x))\le h_0(x)$ by the definition of the join. Inflationarity gives the reverse inequality. Finally, if $p=f(p)$ and $p\ge x$, the lower set $(-\infty ,p]$ is closed by monotonicity of $f$ and the join property. Thus every $h(x)$ is at most $p$, and so is $h_0(x)$. This proves the least fixed point from a directed family of maps: $h_0(x)$ is the least fixed point above $x$.

A topological group is a group equipped with a topology for which multiplication $G\times G\to G$ and inversion $G\to G$ are continuous maps. The product uses the product topology. Translation and inversion are therefore homeomorphisms. Hausdorffness is sometimes included in the definition; the two open-subgroup arguments below do not require it.

For the inverse system, take a directed set $I$, topological spaces $X_i$, and continuous maps $f_{ij}:X_j\to X_i$ for $i\le j$, with $f_{ii}=\mathrm{id}$ and $f_{ik}=f_{ij}{f}_{jk}$. An inverse limit is a space $X$ with compatible continuous maps ${\pi }_i:X\to X_i$ such that every compatible family $q_i:Y\to X_i$ factors uniquely as $q_i={\pi }_{i}q$ through a continuous map $q:Y\to X$. Existence follows from the concrete constructionwith the subspace topology of the product topology. The only possible map $q$ is $q(y)=(q_i(y){)}_i$. Compatibility puts its image in $X$, and coordinatewise continuity makes it continuous. This proves the universal property of an inverse limit, even when $X$ is empty. Given two inverse limits, their universal properties supply maps each way; the composites have the same coordinates as the identity, so uniqueness makes them identities. The inverse limit exists and is unique up to the unique homeomorphism respecting its projections.

A profinite group can be defined as an inverse limit of finite discrete groups, or as a compact Hausdorff space that is a totally disconnected space, equipped with continuous group operations. For the first implication, use the Tychonoff theorem. The compatibility equalities define a closed subgroup of the compact Hausdorff product. Two distinct points are distinguished by a finite discrete coordinate, so no connected subset contains both. Consequently the inverse limit is a compact Hausdorff totally disconnected topological group.

For the converse, use the topological result that a compact Hausdorff totally disconnected space has a basis of clopen sets. Choose a clopen set $U$ containing $1$ inside any specified identity neighbourhood. For each $u\in U$, continuity gives an identity neighbourhood $V_u$ and an open neighbourhood $W_u$ of $u$ with $V_u{W}_u\subseteq U$. Finitely many $W_u$ cover the compact set $U$. Intersect their $V_u$ and its inverse to obtain a symmetric identity neighbourhood $V$ with $VU\subseteq U$. Symmetry also gives $V^{-1}U\subseteq U$, and hence $vU=U$ for every $v\in V$. The setis a subgroup containing $V$, so it is open. It is contained in $U$, since $1\in U$ and $h=h\cdot 1\in hU=U$. By part (ii), $H$ has finite index. Its normal core of a subgroup is a finite intersection of conjugates of $H$, hence an open normal subgroup $N$ contained in $U$. This proves the open normal subgroup basis of a profinite group.

The natural group homomorphismis injective because the intersection of these $N$ is $\{1\}$. For a compatible family of cosets, any finitely many have nonempty intersection: their intersection subgroup is another indexing subgroup, and its prescribed coset lies in each. The finite intersection property and compactness give a point belonging to all the cosets, proving surjectivity. Finally a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. The two definitions of profinite group are equivalent.

For an inverse system of sets $(X_j,f_{ij})$ indexed by a directed set $J$, the inverse limit is the set of compatible tuplesIts projection to $X_j$ sends $(x_i{)}_i$ to $x_j$.

For a directed set of Scott continuous maps $f_i:D\to E$, its least upper bound is $f(x)={\sup }_i{f}_i(x)$. The two directed suprema over $i$ and over inputs commute, proving that $f$ remains Scott continuous. Thus the function space of complete partial orders is a complete partial order.

A monotone map preserving suprema of nonempty directed sets. Between pointed domains it need not send bottom to bottom. The pointwise directed supremum of a directed family of such maps is again Scott continuous: interchange the two directed suprema in ${\sup }_f{\sup }_{x\in A}f(x)$.