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Initial topology

In topology, the initial topology on a set X with respect to a family of functions \{f_i : X \to Y_i\}_{i \in I}, where each Y_i is a topological space, is defined as the coarsest topology on X (i.e., the one with the fewest open sets) that renders all the maps f_i continuous.^[1] This topology is uniquely determined and can be explicitly constructed by taking as a subbasis the collection of all preimages f_i^{-1}(U_i), where U_i is open in Y_i for each i \in I, with the open sets then formed as arbitrary unions of finite intersections of these subbasis elements.^[2] The initial topology is characterized by its universal property: for any topological space Z and any map g: Z \to X, the function g is continuous with respect to the initial topology on X if and only if the composite f_i \circ g: Z \to Y_i is continuous for every i \in I.^[3] This property ensures that the initial topology is the minimal one compatible with the given family of maps, making it a fundamental construction in general topology for inducing structures from existing spaces.^[1] Notable examples include the product topology on a product space \prod_{i \in I} Y_i, which is the initial topology with respect to the projection maps \pi_j: \prod_{i \in I} Y_i \to Y_j for each j \in I, yielding a basis of sets where all but finitely many coordinates are the full spaces.^[1] Similarly, the subspace topology on a subset A \subseteq X of a topological space (X, \tau) is the initial topology induced by the inclusion map i: A \hookrightarrow X, consisting of sets of the form A \cap U for U \in \tau.^[3] In functional analysis, the initial topology plays a key role in defining weak topologies, such as the weak topology on a normed space induced by its dual, which is the coarsest making all continuous linear functionals continuous.^[4] These constructions highlight the initial topology's role in preserving continuity and enabling categorical limits in the category of topological spaces.^[5]

Definition and Construction

Definition

In topology, given a set X and a family of functions \{f_i : X \to Y_i\}_{i \in I} where each Y_i is a topological space, the initial topology on X is the coarsest topology \tau on X such that every f_i is continuous.^[1] A function f: X \to Y between topological spaces is continuous if the preimage f^{-1}(U) of every open set U in Y is open in X.^[1] The coarsest topology is understood to be the one with the fewest open sets among all topologies on X that render the f_i continuous; any strictly coarser topology would fail to make at least one f_i continuous.^[3] This topology is commonly denoted by \tau(\{f_i\}_{i \in I}).^[1] As a special case, the product topology on a product space arises as the initial topology with respect to the family of projection maps.^[1]

Generating subbasis

The initial topology \tau on a set X induced by a family of functions \{f_i : X \to Y_i \mid i \in I\}, where each Y_i is equipped with a topology, is explicitly constructed using a subbasis derived from preimages under these functions. Specifically, the collection \mathcal{S} = \{ f_i^{-1}(U) \mid i \in I, \, U \text{ open in } Y_i \} forms a subbasis for \tau.^[6]^[7]^[8] The topology \tau generated by this subbasis consists of all arbitrary unions of finite intersections of elements from \mathcal{S}. In other words, a subset V \subseteq X belongs to \tau if and only if there exist an index set \alpha and, for each \alpha' \in \alpha, a finite collection of indices i_1, \dots, i_{n_{\alpha'}} \in I and open sets U_{\alpha',1} \subseteq Y_{i_1}, \dots, U_{\alpha',n_{\alpha'}} \subseteq Y_{i_{n_{\alpha'}}} such that

V = \bigcup_{\alpha' \in \alpha} \left( \bigcap_{k=1}^{n_{\alpha'}} f_{i_k}^{-1}(U_{\alpha',k}) \right).

This construction ensures that \mathcal{S} covers X (since X = \bigcup_{i \in I} f_i^{-1}(Y_i) and each Y_i is open in itself), thereby generating a valid topology.^[9]^[8]^[6] To see why \tau is the coarsest topology making all f_i continuous, note that any topology \sigma on X for which each f_i: (X, \sigma) \to (Y_i, \text{its topology}) is continuous must contain all sets in \mathcal{S}, since continuity requires f_i^{-1}(U) to be \sigma-open for every open U \subseteq Y_i. Thus, \tau, being the smallest topology containing \mathcal{S} (as the intersection of all topologies containing \mathcal{S}), is coarser than or equal to \sigma.^[7]^[9]^[6] Finally, each f_i is continuous with respect to \tau because the subbasis elements f_i^{-1}(U) are open in \tau by construction, and finite intersections of such preimages under f_i remain preimages of open sets in Y_i. Moreover, \tau is the smallest such topology, as any coarser topology would fail to include some elements of \mathcal{S}, violating continuity of at least one f_j.^[7]^[8]^[6]

Examples

Subspace and product topologies

In topology, the subspace topology provides a fundamental example of an initial topology. Consider a topological space Y with topology \mathcal{T}_Y and a subset A \subseteq Y. The initial topology on A induced by the inclusion map \iota: A \to Y is the coarsest topology on A that makes \iota continuous. This topology, known as the subspace topology, has as its subbasis the collection \{\iota^{-1}(U) \mid U \in \mathcal{T}_Y\} = \{U \cap A \mid U \in \mathcal{T}_Y\}. Consequently, the open sets in A are precisely the intersections of open sets in Y with A.^[10]^[5] A concrete illustration arises in the real line \mathbb{R} with its standard topology. For the subspace A = [0,1] \subseteq \mathbb{R}, open sets include (0,1) \cap [0,1] = (0,1), which remains open in the subspace, while sets like (-1,0.5) \cap [0,1] = [0,0.5) are also open in [0,1]. This structure ensures that the subspace topology restricts the openness from the ambient space appropriately, preserving continuity of the inclusion.^[10] The product topology similarly exemplifies the initial topology for families of spaces. Given a family of topological spaces \{X_i\}_{i \in I} with topologies \mathcal{T}_i, the product space is \prod_{i \in I} X_i, and the initial topology on it is induced by the family of projection maps \pi_j: \prod_{i \in I} X_i \to X_j for each j \in I. This is the coarsest topology making all \pi_j continuous, with subbasis \{\pi_j^{-1}(U_j) \mid j \in I, U_j \in \mathcal{T}_j\}, consisting of cylinder sets that are open in all but one coordinate. The basis for this topology comprises finite intersections of these subbasis elements.^[10]^[5] For the finite product \mathbb{R}^2 = \mathbb{R} \times \mathbb{R}, the product topology has basis elements that are open rectangles (a,b) \times (c,d), where a < b, c < d, aligning with the standard Euclidean topology on the plane. This construction generalizes to arbitrary index sets I, where openness is determined by finitely many coordinates, ensuring compatibility with the projections.^[10] Thus, the subspace topology is the initial topology with respect to a single inclusion map, while the product topology is the initial topology with respect to the family of projection maps.^[10]^[5]

Topologies on function spaces

In the context of function spaces, the initial topology plays a central role in defining convergence structures that align with natural notions in analysis. One prominent example is the pointwise convergence topology on the space Y^X of all functions from a topological space X to a topological space Y, which is the initial topology induced by the family of evaluation maps \mathrm{ev}_x: Y^X \to Y, f \mapsto f(x), for each x \in X.^[11] The subbasis for this topology consists of sets of the form \mathrm{ev}_x^{-1}(U) = \{ f \in Y^X \mid f(x) \in U \}, where U is open in Y.^[11] This structure coincides with the product topology on Y^X, viewed as the product of |X| copies of Y, ensuring that nets (or sequences, if applicable) converge pointwise if and only if they converge in this topology.^[11] When restricting to the subspace C(X, Y) of continuous functions, the pointwise convergence topology is obtained as the subspace topology from Y^X, again generated as the initial topology with respect to the evaluation maps \mathrm{ev}_x for x \in X.^[11] This topology is particularly useful in analysis for studying pointwise limits of continuous functions, though it may not preserve continuity in general. Another important instance arises in the topology of uniform convergence, often considered on spaces of bounded or continuous functions equipped with a uniform structure. For the space C_b(X) of bounded continuous real-valued functions on a uniform space X, the uniform convergence topology is the initial topology generated by the family of seminorms p(f) = \sup_{x \in X} |f(x)|, or more generally, by seminorms p_K(f) = \sup_{x \in K} |f(x)| over bounded subsets K \subseteq X.^[12] This yields the coarsest locally convex topology making all these seminorms continuous, corresponding to uniform convergence on bounded sets.^[12] In cases where X is compact, this reduces to the sup-norm topology on C(X). A concrete illustration occurs on the space \mathbb{R}^\mathbb{R} of all real-valued functions on \mathbb{R}, where the pointwise convergence topology has a subbasis consisting of sets where functions take values in specified open intervals at finitely many points in \mathbb{R}; basic open neighborhoods thus constrain agreement on finite subsets within opens.^[11] In contrast, the compact-open topology on C(X, Y) is the initial topology with respect to the evaluation maps \mathrm{ev}_K: C(X, Y) \to Y^K, f \mapsto f|_K, for compact subsets K \subseteq X, where Y^K carries the product topology; its subbasis comprises sets \{ f \in C(X, Y) \mid f(K) \subseteq U \} for compact K and open U \subseteq Y.^[13] This finer structure captures uniform convergence on compacts, distinguishing it from the coarser pointwise topology.

Weak topologies and inverse limits

In functional analysis, the weak topology on a topological vector space V over \mathbb{R} or \mathbb{C} is defined as the initial topology induced by the family of all continuous linear functionals \phi: V \to \mathbb{R} (or \mathbb{C}), where the dual space V^* consists of these functionals.^[14] This topology has a subbasis consisting of sets of the form \phi^{-1}(U), where \phi \in V^* and U is open in the scalar field.^[15] The resulting topology is the coarsest one making all elements of V^* continuous, and it separates points if and only if V^* separates points in V. A concrete example arises in sequence spaces such as \ell^p for $1 \leq p < \infty, where the weak topology is generated by the dual pairings with \ell^q (with $1/p + 1/q = 1).^[16] Here, the subbasis elements are \{ x \in \ell^p : |\langle x, y \rangle| < \epsilon \} for y \in \ell^q and \epsilon > 0, reflecting the initial topology from the evaluation maps. This weak topology on \ell^p is strictly coarser than the norm topology and ensures point separation due to the reflexivity of \ell^p.^[17] In the context of inverse limits, consider a projective system of topological spaces \{X_i, p_{ij}: X_i \to X_j\}_{i \geq j} indexed by a directed set I. The inverse limit \lim_{\leftarrow} X_i is the set of compatible threads \{(x_i)_{i \in I} : p_{ij}(x_i) = x_j \text{ for } i \geq j\}, equipped with the initial topology induced by the projection maps \pi_k: \lim_{\leftarrow} X_i \to X_k for each k \in I.^[18] This topology coincides with the subspace topology inherited from the product space \prod_{i \in I} X_i under the product topology, making the projections \pi_k continuous by construction.^[19] A prominent concrete realization is the ring of p-adic integers \mathbb{Z}_p, which forms the inverse limit \lim_{\leftarrow} \mathbb{Z}/p^n \mathbb{Z} with transition maps given by the natural projections \mathbb{Z}/p^{n+1} \mathbb{Z} \to \mathbb{Z}/p^n \mathbb{Z}.^[20] Each factor \mathbb{Z}/p^n \mathbb{Z} carries the discrete topology, so the initial topology on \mathbb{Z}_p—induced by these projections—is the coarsest making all \pi_n: \mathbb{Z}_p \to \mathbb{Z}/p^n \mathbb{Z} continuous, resulting in a compact, totally disconnected topological ring.^[21] This structure underpins the p-adic topology, where basic open sets are the kernels of these projections.^[22]

Basic Properties

Characteristic property

The initial topology \tau on a set X induced by a family of continuous maps \{f_i : X \to Y_i\}_{i \in I}, where each (Y_i, \mathcal{T}_i) is a topological space, is characterized by a universal property that distinguishes it uniquely among all possible topologies on X. Specifically, \tau is the initial topology with respect to \{f_i\} if and only if, for any topological space (Z, \rho) and any map g: Z \to X, the map g is continuous from (Z, \rho) to (X, \tau) precisely when each composition f_i \circ g: Z \to Y_i is continuous from (Z, \rho) to (Y_i, \mathcal{T}_i) for all i \in I.^[6]^[23] This property arises from the construction of [\tau](/page/Tau) as the coarsest topology making all f_i continuous, ensuring that continuity with respect to [\tau](/page/Tau) is equivalent to preserving the preimages of open sets in each Y_i. To see one direction, if g is (\rho, [\tau](/page/Tau))-continuous, then for each i, the composition f_i \circ g is continuous as a composition of continuous maps, since each f_i is (\tau, \mathcal{T}_i)-continuous by definition of [\tau](/page/Tau). Conversely, suppose each f_i \circ g is continuous; then for any subbasic open set U = f_j^{-1}(V_j) in the subbasis generating [\tau](/page/Tau) (with V_j \in \mathcal{T}_j), the preimage g^{-1}(U) = (f_j \circ g)^{-1}(V_j) is open in \rho, so g^{-1} preserves the subbasis and hence all of [\tau](/page/Tau), making g continuous.^[6] The universal property implies that \tau is the unique coarsest such topology: any coarser topology \sigma \subsetneq \tau would fail to make at least one f_i continuous, violating the condition, while any finer topology \tau' \supset \tau would still satisfy the continuity equivalences but is not the minimal one. Thus, any two topologies satisfying this property must coincide, as the coarser of the two would contradict the minimality implied by the universal characterization.^[6]^[23]

Continuity of evaluation maps

In the initial topology \tau on a set X induced by a family of maps \{f_i : X \to Y_i\}_{i \in I} to topological spaces (Y_i, \tau_i), each map f_i : (X, \tau) \to (Y_i, \tau_i) is continuous by the defining property of the initial topology, as \tau is the coarsest topology making all such maps continuous.^[9] This follows directly from the construction of \tau, where the subbasis consists of sets f_i^{-1}(U_i) for open U_i \in \tau_i, ensuring that preimages under each f_i of open sets in Y_i are open in X.^[9] For function spaces, consider the set Y^X of all functions from a topological space (X, \tau_X) to a topological space (Y, \tau_Y). The pointwise topology (also called the topology of pointwise convergence) on Y^X is the initial topology induced by the family of evaluation maps \{\mathrm{ev}_x : Y^X \to Y\}_{x \in X}, where \mathrm{ev}_x(f) = f(x).^[24] By construction, each individual evaluation map \mathrm{ev}_x : (Y^X, \tau_{\mathrm{pt}}) \to (Y, \tau_Y) is continuous, as the subbasis for \tau_{\mathrm{pt}} comprises sets of the form \mathrm{ev}_x^{-1}(U) = \{f \in Y^X \mid f(x) \in U\} for open U \in \tau_Y.^[24] The joint evaluation map \mathrm{ev} : Y^X \times X \to Y, defined by \mathrm{ev}(f, x) = f(x), is continuous when Y^X \times X is equipped with the product topology \tau_{\mathrm{pt}} \times \tau_X if and only if X is discrete.^[25] In contrast, joint continuity of \mathrm{ev} holds in the finer compact-open topology on Y^X (initial with respect to maps \mathrm{ev}_K : Y^X \to Y^K for compact K \subseteq X) when X is locally compact Hausdorff, ensuring preimages align with compact neighborhoods.^[26]

Transitivity

The transitivity of initial topologies refers to the property that these topologies compose naturally under compositions of functions, allowing hierarchical constructions to preserve the initial structure. Specifically, suppose a set X is equipped with the initial topology \tau induced by a family of maps \{f_i : X \to Y_i\}_{i \in I}, where each Y_i carries the initial topology \sigma_i induced by a family \{g_{ij} : Y_i \to Z_{ij}\}_{j \in J_i} and each Z_{ij} is a topological space. Then \tau coincides with the initial topology on X induced by the composed family \{g_{ij} \circ f_i : X \to Z_{ij}\}_{(i,j) \in I \times J_i}.^[27] This theorem follows from the characteristic (universal) property of initial topologies: a map h : W \to X from a topological space W is continuous with respect to \tau if and only if f_i \circ h : W \to Y_i is continuous for all i \in I. For the composed family, continuity of h requires continuity of each g_{ij} \circ f_i \circ h. Since \sigma_i is initial on Y_i, the map f_i \circ h : W \to Y_i is continuous if and only if g_{ij} \circ (f_i \circ h) is continuous for all j \in J_i, which aligns precisely with the continuity conditions for the original family \{f_i\}. Thus, the universal property holds equivalently for both families, establishing that \tau is initial with respect to the compositions.^[27] The coarseness of \tau is preserved because the subbasis generating \tau consists of sets of the form f_i^{-1}(U_i) for U_i open in Y_i. Under \sigma_i, the subbasis for Y_i is generated by \{g_{ij}^{-1}(V_{ij}) : V_{ij} \in \mathcal{T}_{Z_{ij}}\}_{j \in J_i}, so

f_i^{-1}(U_i) = f_i^{-1}\left( \bigcup \text{ finite intersections of } g_{ij}^{-1}(V_{ij}) \right) = \bigcup \text{ finite intersections of } (g_{ij} \circ f_i)^{-1}(V_{ij}),

showing that the subbasis from the composed family generates exactly the same topology as the original.^[27] A key application of this transitivity is in product topologies: the product topology on \prod_{i \in I} X_i can be viewed as initial with respect to the projections \pi_i : \prod X_i \to X_i, and if each X_i = \prod_{j \in J_i} X_{ij}, then the projections compose with those on the inner products, yielding by transitivity the initial topology with respect to all ultimate projections \pi_{ij} : \prod_{i \in I} \left( \prod_{j \in J_i} X_{ij} \right) \to X_{ij}. This confirms that iterated products coincide with the overall product topology.

Separation Properties

Hausdorffness criteria

The initial topology \tau on a set X induced by a family of continuous maps \{f_i: X \to Y_i \mid i \in I\}, where each (Y_i, \mathcal{T}_i) is a topological space, is Hausdorff if and only if each Y_i is Hausdorff and the family \{f_i\} separates points on X, meaning that for all distinct x, y \in X, there exists some i \in I such that f_i(x) \neq f_i(y).^[28] To see the necessity of the separation condition, suppose each Y_i is Hausdorff but the family fails to separate some distinct x, y \in X, so f_i(x) = f_i(y) for all i. Then any subbasic open set in \tau, which is of the form f_j^{-1}(U_j) for open U_j \subseteq Y_j, either contains both x and y or neither, since f_j(x) = f_j(y). Finite intersections of such subbasic sets thus also fail to separate x and y, so no basic open sets separate them, implying \tau cannot be Hausdorff.^[29]^[28] For the sufficiency, assume each Y_i is Hausdorff and the family separates points. Let x, y \in X be distinct; then there exists i with f_i(x) \neq f_i(y). Since Y_i is Hausdorff, there exist disjoint open sets U_i, V_i \subseteq Y_i such that f_i(x) \in U_i and f_i(y) \in V_i. The preimages f_i^{-1}(U_i) and f_i^{-1}(V_i) are then open in \tau (as subbasic sets) and disjoint, with x \in f_i^{-1}(U_i) and y \in f_i^{-1}(V_i), so \tau is Hausdorff.^[28] A concrete example arises in the product topology on \prod_{i \in I} X_i, which is the initial topology induced by the projection maps \pi_j: \prod_{i \in I} X_i \to X_j for each j \in I. Here, the projections jointly separate points because if (x_i)_{i \in I} \neq (y_i)_{i \in I}, then x_k \neq y_k for some k, so \pi_k((x_i)) \neq \pi_k((y_i)). Thus, if each X_j is Hausdorff, the product is Hausdorff.^[9]

Separation of points from closed sets

In the initial topology \tau on X induced by \{f_i : X \to Y_i \mid i \in I\}, a point x \in X lies in the closure \overline{C}^\tau of a subset C \subseteq X if and only if f_i(x) \in \overline{f_i(C)}^{Y_i} for every i \in I. Equivalently, x \notin \overline{C}^\tau if and only if there exists i \in I such that f_i(x) \notin \overline{f_i(C)}^{Y_i}.^[30] If C is closed in \tau, then \overline{C}^\tau = C, so for x \notin C, there always exists an open U \ni x with U \cap C = \emptyset; by the characterization, this holds if and only if there exists i with f_i(x) \notin \overline{f_i(C)}^{Y_i}. To see the forward direction, suppose there exists i such that f_i(x) \notin \overline{f_i(C)}^{Y_i}. Then there is open V_i \ni f_i(x) in Y_i with V_i \cap f_i(C) = \emptyset. The set U = f_i^{-1}(V_i) is open in \tau, contains x, and U \cap C \subseteq f_i^{-1}(V_i \cap f_i(C)) = \emptyset. For the converse, suppose for every i, f_i(x) \in \overline{f_i(C)}^{Y_i}. To show x \in \overline{C}^\tau, consider any basic open neighborhood B = \bigcap_{k=1}^n f_{i_k}^{-1}(V_k) \ni x in \tau, where each V_k \ni f_{i_k}(x) is open in Y_{i_k}. Consider the joint map g: X \to \prod_{k=1}^n Y_{i_k} given by g(z) = (f_{i_1}(z), \dots, f_{i_n}(z)). Then B = g^{-1}(\prod_{k=1}^n V_k), and \prod V_k \ni g(x) is open in the product topology, which is initial with respect to the projections \pi_k. By the closure characterization in the product (applied recursively or by the same logic), since \pi_k(g(x)) = f_{i_k}(x) \in \overline{f_{i_k}(C)}^{Y_{i_k}} = \overline{\pi_k(g(C))} for each k, it follows that g(x) \in \overline{g(C)} in the product, so \prod V_k \cap g(C) \neq \emptyset, hence B \cap C \neq \emptyset. Thus, every neighborhood of x intersects C, so x \in \overline{C}^\tau.^[30] If each Y_i is regular and \tau is T_0, then (X, \tau) is regular. For x \in X and closed C \subseteq X with x \notin C, the closure characterization yields i with f_i(x) \notin \overline{f_i(C)}^{Y_i}. By regularity of Y_i, there exist disjoint open sets V_i \ni f_i(x) and W_i \supseteq \overline{f_i(C)}^{Y_i} in Y_i. Then U = f_i^{-1}(V_i) and V = f_i^{-1}(W_i) are disjoint open sets in \tau separating x from C, since x \in U and C \subseteq f_i^{-1}(\overline{f_i(C)}^{Y_i}) \subseteq V.^[30] A concrete instance arises in the subspace topology, which is the initial topology induced by the inclusion map i: S \hookrightarrow X from a subset S \subseteq X to a regular space (X, \mathcal{T}). Subspaces of regular spaces are regular: for p \in S and C \subseteq S closed in S (so \overline{i(C)}^X \cap S = i(C)), p \notin C implies i(p) \notin \overline{i(C)}^X, and regularity of X provides disjoint opens U' \ni i(p), V' \supseteq \overline{i(C)}^X in X. Then U = U' \cap S \ni p and V = V' \cap S \supseteq C are disjoint opens in S.^[31]

Advanced Structures

Initial uniform structure

In the context of uniform spaces, the initial uniform structure provides a canonical way to equip a set with a uniformity based on a family of maps to known uniform spaces. Specifically, given a set X and a family of maps \{f_i : X \to Y_i\}_{i \in I}, where each (Y_i, \mathcal{U}_i) is a uniform space with uniformity \mathcal{U}_i, the initial uniform structure \mathcal{U} on X is defined as the coarsest uniformity such that every f_i is uniformly continuous.^[27] This structure ensures that uniform continuity of the f_i is preserved while being as weak as possible, analogous to the initial topology in the topological setting. The construction of \mathcal{U} proceeds by generating its basis from preimages under the maps f_i. A basis for the entourages consists of sets of the form

W = \bigcap_{k=1}^n (f_{i_k} \times f_{i_k})^{-1}(V_{i_k}),

where n is finite, each i_k \in I, and each V_{i_k} \in \mathcal{U}_{i_k} is an entourage in Y_{i_k}. These basic entourages capture the "closeness" relations pulled back from the target spaces, satisfying the axioms of a uniformity (reflexivity, symmetry, and transitivity) by construction.^[27] Any other uniformity on X making all f_i uniformly continuous must be finer than \mathcal{U}, establishing its universality. The initial uniform structure \mathcal{U} induces a topology on X via the standard association between uniformities and topologies, where neighborhoods of a point are generated from entourage intersections. This induced topology coincides precisely with the initial topology on X with respect to the family \{f_i\}, confirming compatibility between the uniform and topological initial constructions.^[27] Furthermore, when viewed in the category of uniform spaces, the uniform topology generated by \mathcal{U} is initial with respect to the f_i as uniformly continuous maps. A prominent example arises in function spaces. Consider the set Y^X of all functions from a set X to a uniform space (Y, \mathcal{U}_Y). The uniformity of uniform convergence on Y^X is the initial uniform structure induced by the evaluation maps \mathrm{ev}_x : Y^X \to Y defined by \mathrm{ev}_x(f) = f(x) for each x \in X. The basic entourages are then of the form \{(f, g) \in Y^X \times Y^X \mid (f(x), g(x)) \in V \ \forall x \in X\} for V \in \mathcal{U}_Y. If Y is metrizable with metric d, this uniformity corresponds to the supremum pseudometric d_u(f, g) = \sup_{x \in X} d(f(x), g(x)), which measures uniform closeness across the domain.^[27]

Categorical description

In the category Top of topological spaces and continuous maps, the initial topology on a set X with respect to a family of maps \{f_i : X \to Y_i\}_{i \in I} into topological spaces \{(Y_i, \tau_i)\}_{i \in I} is the unique topology \tau on X such that the object (X, \tau) is initial in the comma category (\mathbf{Top} \downarrow \prod_{i \in I} Y_i)_{f_i}, where the structure is given by the maps f_i.^[24] This means that (X, \tau) is universal in the sense that for any topological space Z and family of maps \{h_i : Z \to Y_i\}_{i \in I}, there exists a unique continuous map g : Z \to (X, \tau) such that h_i = f_i \circ g for all i \in I, provided such a g exists on the underlying sets.^[24] The universal property can be restated hom-set wise: the set of continuous maps \mathrm{Hom}_{\mathbf{Top}}((Z, \sigma), (X, \tau)) is naturally isomorphic to the set of families \{(h_i : Z \to Y_i)_{i \in I} \mid h_i = f_i \circ g for some continuous g : (Z, \sigma) \to (X, \tau)\}.^[24] This isomorphism captures the initiality by ensuring that the topology \tau is the coarsest one compatible with the f_i, as any coarser topology would fail the uniqueness or existence of such g. This categorical perspective unifies the initial topology with limits in Top: the product topology on \prod_{i \in I} Y_i arises as the categorical product, equipped with the initial topology relative to the projections \pi_i : \prod Y_i \to Y_i; the subspace topology on a subset A \subseteq X is the pullback along the inclusion A \hookrightarrow X; and the inverse limit of an inverse system of spaces is the projective limit in Top, again with the initial topology induced by the bonding maps.^[24] Although Top admits all small limits via initial topologies on underlying set-limits, it requires final topologies for colimits (such as quotients), highlighting that initial structures suffice for limits but not the full cocompleteness without their duals.^[24] In contrast, the final topology on X with respect to a family of maps out of X makes (X, \tau) terminal in the opposite comma category, dualizing the role of initiality.^[24]

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Jul 29, 2016 · (Initial topology) Given a map f : X ! Y and a topology on Y , let T be the collection of subsets of X of the form fL1(U) where U is open ...
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[PDF] Topology (H) Lecture 5
Mar 22, 2021 · The F-induced topology (which is also known as initial topology or weak topology or limit topology)on X, denoted by TF , is defined to be ...
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[PDF] 06. Initial and final topology - TU Graz
The initial topology on A with respect to the inclusion function j : A → X where j(x) = x ∀ x ∈ A , is called the subspace topology on A and denoted by τ|A .
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[PDF] Initial and final topology - msleziak.com
Oct 17, 2024 · ... I,U ∈ Si } is also a subbase for the initial topology. Initial and nal topology. Page 4. Initial topology. Examples. Examples of initial ...
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[PDF] An Introduction to Topology
The initial topology τ defined by the family fi : X → Yi, i ∈ I, of maps is the coarsest topology for which all maps fi are continuous. It has the.
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[PDF] [DRAFT] A Peripatetic Course in Algebraic Topology
Jul 22, 2016 · The product topology is the initial topology with respect to the projection maps (the underlying space is the set product). (TODO: diagram).
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[PDF] Function Spaces
Our next task is to define a topology on Y X under which convergence of sequences corresponds to pointwise convergence of functions. Definition: The Product ...Missing: initial | Show results with:initial
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[PDF] Notes on Distributions - Math (Princeton)
The l.c. topology on Cm(Ω) is generated by the seminorms. pK,m as K ⊂⊂ Ω runs through all compact subsets of Ω. This is the topology of uniform convergence ...
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[PDF] GENERAL TOPOLOGY Tammo tom Dieck - Mathematisches Institut
Oct 2, 2018 · initial topology with respect to the fj . An example of an ... In this section we study the compact-open topology on mapping spaces.
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[PDF] Functional Analysis Princeton University MAT520 Lecture Notes
Aug 18, 2023 · ... initial topology generated by X∗∗ whereas the weak-star topology is characterized by the initial topology generated by J (X). Actually the ...
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[PDF] Introduction to functional analysis - MIT Mathematics
May 21, 2025 · In the context of topological vector spaces, the initial topology is also called weak topology. ... The weak topology on X is the initial topology.
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[PDF] Functional Analysis - Penn Math
Oct 4, 2016 · 3.2 Lp spaces . ... The weak topology on X generated by ¿, denoted by σ(X,¿), is the smallest topology on X which makes each f ∈ ¿ be ...
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[PDF] Functional Analysis - Lecture Notes - UC Berkeley math
2.1 Dual space of Lp . ... The product topology on X is the weak topology σ(X,{πγ | γ ∈ Γ}). So V ⊆ X is open iff for all x ∈ V there exist n ∈ N, γ1 ...
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[PDF] Geometric Group Theory - UC Davis Math
... initial topology, i.e. the subset topology of the Tychonoff topology on the ... subspace topology. Then we have a natural continuous embedding ι : Y i ...
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[PDF] Infinite-Dimensional Diagonalization and Semisimplicity - UCCS
... inverse limit of the Mj as R- modules and the inverse limit as topological groups.) The topology on the inverse limit is the initial topology with respect ...
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[PDF] 8 Complete fields and valuation rings
Oct 4, 2016 · topology of the inverse limit X := lim. ←−n. A/πnA. The valuation ... the ring of p-adic integers. Example 8.14. Let K = Fq(t) be the ...
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[PDF] the p-adic integers, analytically and algebraically - UChicago Math
Aug 26, 2018 · The inverse limit definition of the p-adics is equivalent to the Cauchy comple- tion of Z under the p-adic norm. All three definitions of the p- ...
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[PDF] INVERSE LIMITS AND PROFINITE GROUPS - OSU Math
Inverse limits. Standard examples of inverse limits arise from sequences of groups, with maps between them: for instance, if we have the sequence Gn = Z/pn.
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[PDF] Unique Lifting to a Functor
Then we define the under category or comma category (X ↓ C) of objects under X by setting ... the initial topology. Exactly the same argument holds for ...
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[PDF] Topology for the working mathematician - Mathematics
Jan 9, 2019 · ... initial topology ... General Topology', also called 'set-theoretic topology' or 'point-set topology', which provides the foundations for ...
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[PDF] Introduction to Topology -- 1 in nLab
Aug 9, 2017 · whose topology is the initial topology, def. 6.17, for the ... John Kelley General Topology, Graduate Texts in Mathematics, Springer (1955).
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[PDF] General Topology - Fakultät für Mathematik - Universität Wien
Jan 10, 2019 · (i) Some authors, (e.g., Bourbaki) call the above property quasi- ... initial topology with respect to (pn ◦ ϕ)n∈N, 1.2.3 implies that O ...
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[PDF] Functional Analysis, Math 7320 Lecture Notes from November 15 ...
Nov 15, 2016 · The F-topology is Hausdorff if and only if F separates the points of X. ... Denote by τ0 the initial topology induced by V 0 on V , then (V ...
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Hausdorff property with initial topology - Math Stack Exchange
Jul 28, 2022 · Consider a topological space X such that the topology on X is the initial topology with respect to the family of maps fi:X→Yi where Yi are ...Hausdorff spaces and separated maps. - Math Stack ExchangeIf a function space is Hausdorff, then is the codomain of functions ...More results from math.stackexchange.comMissing: codomains | Show results with:codomains
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The Stone-Čech compactification of Tychonoff spaces - Jordan Bell
Jun 27, 2014 · This shows that C b ⁢ ( X ) separates points from closed sets in X . Applying Theorem 5, we get that X has the initial topology induced by ...
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[PDF] Lemma 31.1 - Introduction to Topology
Aug 12, 2016 · Hausdorff spaces is Hausdorff. (b) A subspace of a regular space is regular. A product of regular spaces is regular. Proof. (a) Let X be ...