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Category of topological spaces

The category of topological spaces, commonly denoted Top, consists of all topological spaces as objects and continuous functions between them as morphisms. This structure formalizes the study of continuity and topological invariants within the framework of , providing a unified way to analyze spaces through their mappings rather than intrinsic properties alone. Top is both complete and cocomplete, possessing all small limits and colimits, which correspond to familiar topological constructions such as pullbacks, equalizers, pushouts, and coequalizers. Finite products in Top are given by the equipped with the , where a into the product is continuous if and only if its projections are continuous. Coproducts, or s, are formed by the set-theoretic with the making each component an open , ensuring the maps are both open and continuous embeddings. These limits and colimits enable the categorical treatment of gluing and constructions central to . Unlike the , Top is not Cartesian closed, meaning internal function spaces do not generally exist as topological spaces in a way that satisfies the universal property for exponentials. This limitation has motivated the development of subcategories, such as compactly generated Hausdorff spaces, that are Cartesian closed and better suited for and . Nonetheless, Top remains a example in , illustrating how abstract categorical tools can illuminate concrete mathematical disciplines like and .

Definition and objects

Objects as topological spaces

In the category of topological spaces, known as Top, the objects are precisely the topological spaces. A topological space is a pair (X, \tau), where X is a set and \tau is a of the power set \mathcal{P}(X) satisfying the following axioms: the \emptyset and X belong to \tau; \tau is closed under arbitrary unions; and \tau is closed under finite intersections. This axiomatic framework, formalized by , generalizes classical notions of space by abstracting the properties essential for defining without relying on metrics or distances. Key examples illustrate the versatility of this structure. The \mathbb{R}^n, equipped with the standard topology, has as its open sets arbitrary unions of open balls \{ y \in \mathbb{R}^n \mid \| y - x \| < r \} for x \in \mathbb{R}^n and r > 0. On any set X, the discrete topology takes \tau = \mathcal{P}(X), rendering every subset open and maximizing the number of open sets. Conversely, the indiscrete (or on X restricts \tau = \{\emptyset, X\}, minimizing the open sets to only the and the whole space. These objects in embody generalized spatial structures where topological properties such as openness enable the uniform treatment of continuity across diverse settings, from metric approximations like spaces to purely set-theoretic extremes like and indiscrete cases.

Morphisms as continuous functions

In the category of topological spaces, denoted , the morphisms are continuous functions between topological spaces. A function f: X \to Y between topological spaces X and Y is continuous if for every U \subseteq Y, the preimage f^{-1}(U) \subseteq X is open. This definition ensures that the categorical arrows respect the topological structure by preserving openness under inverse images. Composition of morphisms in is given by the standard composition of functions, which preserves continuity: if f: X \to Y and g: Y \to Z are continuous, then g \circ f: X \to Z is continuous. The identity morphism for any object X is the identity function \mathrm{id}_X: X \to X, which is continuous since the preimage of any open set U \subseteq X under \mathrm{id}_X is U itself. These properties establish as a category where the morphisms form a category under function composition. Examples of morphisms include inclusion maps between subspaces. For a subspace S \subseteq X equipped with the subspace topology, the inclusion \iota_S: S \hookrightarrow X is continuous because the preimage under \iota_S of an open set in X intersects S in an open set relative to S. Homeomorphisms serve as isomorphisms in Top: a bijective continuous function f: X \to Y with continuous inverse f^{-1}: Y \to X is an isomorphism, as both f and f^{-1} preserve the topological structure.

Concrete structure over Set

Forgetful functor

The forgetful functor U: \mathbf{Top} \to \mathbf{Set}, also denoted G in some texts, maps each topological space (X, \tau) in \mathbf{Top} to its underlying set X in \mathbf{Set}, thereby discarding the topological structure \tau. It sends each continuous function f: (X, \tau) \to (Y, \sigma) to the corresponding function f: X \to Y between sets, preserving the action on morphisms without regard to continuity. This construction is standard in category theory, reflecting the underlying set as the "carrier" of the topological structure. The U is faithful, meaning it induces injective maps on hom-sets: U_{X,Y}: \mathbf{Top}(X,Y) \to \mathbf{Set}(X,Y) is for any objects X, Y, since distinct continuous functions differ as set functions. However, U is not full, as its action on hom-sets is not surjective; for instance, not every between the underlying sets of two topological spaces is continuous with respect to their topologies. These properties highlight how U embeds the morphisms of \mathbf{Top} strictly into those of \mathbf{Set}, capturing distinctions but omitting the full range of set-theoretic maps. Via the faithful functor U, the category \mathbf{Top} is concrete over \mathbf{Set}, meaning its objects can be represented as sets equipped with additional structure (here, a topology), and its morphisms as structure-preserving maps between them. Concreteness facilitates the study of \mathbf{Top} by leveraging the simpler framework of \mathbf{Set}, such as in constructing limits or analyzing embeddings, while ensuring that the representation distinguishes distinct morphisms. This perspective underscores \mathbf{Top} as a structured extension of \mathbf{Set}, where the forgetful functor provides a canonical way to recover the base category.

Underlying set preservation

The forgetful functor U: \mathbf{Top} \to \mathbf{Set} maps each topological space to its underlying set, thereby discarding the topological structure while retaining the set-theoretic foundation. This functor is faithful and preserves the underlying sets in key categorical operations, such as products and coproducts, allowing topological constructions to align directly with their set-theoretic counterparts at the level of underlying sets. For products, the categorical product in \mathbf{[Top](/page/Top)} of two topological spaces X and Y is the set X \times Y equipped with the , where the underlying set is exactly the of the underlying sets U(X) \times U(Y). Thus, applying the yields U(X \times Y) \cong U(X) \times U(Y), ensuring that the set-theoretic product structure is preserved without alteration. This property holds more generally for arbitrary products in \mathbf{[Top](/page/Top)}, reflecting the concrete nature of the category over \mathbf{Set}. In the case of coproducts, the coproduct in \mathbf{Top} of X and Y is the disjoint union X \sqcup Y with the disjoint union (or sum) topology, and its underlying set is the disjoint union of the underlying sets U(X) \sqcup U(Y). Consequently, U(X \sqcup Y) \cong U(X) \sqcup U(Y), mirroring the coproduct in \mathbf{Set}. This preservation extends to infinite coproducts as well, underscoring the functor's role in maintaining set-level colimits. These preservation properties enable the lifting of set-theoretic constructions—such as forming products or coproducts of sets—to the category , where the underlying sets remain unchanged but are endowed with topologies that guarantee the continuity of the induced morphisms. However, this lifting requires careful adjustment of the topology to satisfy conditions, which can restrict the morphisms compared to the freer structure in \mathbf{Set}, highlighting the interplay between algebraic and topological aspects in categories.

Limits and colimits

Finite limits

The category of topological spaces, denoted Top, possesses all finite limits, making it a finitely complete category. These limits are constructed concretely as subspaces of products in the underlying , endowed with the appropriate topologies to ensure the universal property holds with respect to continuous functions. This construction aligns with the general that limits in Top are formed by taking the and equipping it with the with respect to the maps. Finite products in are given by the of the underlying sets, equipped with the . For a finite family of topological spaces \{X_i\}_{i \in I}, the product \prod_{i \in I} X_i has as its topology the coarsest topology making all projection maps \pi_j: \prod_{i \in I} X_i \to X_j continuous for j \in I. A basis for this topology consists of sets of the form \prod_{i \in I} U_i, where each U_i is open in X_i and U_i = X_i for all but finitely many i. This ensures that the product satisfies the universal : for any Y with continuous maps f_i: Y \to X_i, there exists a unique continuous f: Y \to \prod X_i such that \pi_i \circ f = f_i for all i. Binary products, such as X \times Y, follow the same construction and serve as building blocks for higher finite products. Equalizers in arise as subspaces of a given space where two parallel continuous maps agree. Specifically, for continuous functions f, g: X \to Y, the equalizer \operatorname{Eq}(f, g) is the subspace \{x \in X \mid f(x) = g(x)\} of X, endowed with the induced from X. The e: \operatorname{Eq}(f, g) \hookrightarrow X is continuous and satisfies the universal property: for any space Z with a continuous h: Z \to X such that f \circ h = g \circ h, there is a unique continuous \overline{h}: Z \to \operatorname{Eq}(f, g) with e \circ \overline{h} = h. This construction preserves the topological structure, as the subspace topology ensures continuity of the mediating maps. Equalizers are absolute, meaning they are preserved by any , including the to Set. Pullbacks, or fiber products, in can be constructed using products and equalizers. For a diagram X \xrightarrow{f} Z \xleftarrow{g} Y, the pullback is the P = \{(x, y) \in X \times Y \mid f(x) = g(y)\} of the product X \times Y, equipped with the . The projections \pi_X: P \to X and \pi_Y: P \to Y are continuous, and P satisfies the universal property: for any W with continuous u: W \to X and v: W \to Y such that f \circ u = g \circ v, there exists a unique continuous w: W \to P with \pi_X \circ w = u and \pi_Y \circ w = v. This inherits openness and closedness properties from the , ensuring compatibility with continuous functions. In , pullbacks are stable under pullback, a key feature for categorical constructions in topology. A representative example of a pullback in Top occurs when considering solution sets to topological equations. For instance, given continuous maps f: X \to \mathbb{R} and g: Y \to \mathbb{R}, the along f and g yields the space of pairs (x, y) where f(x) = g(y), which topologically represents the to f(x) - g(y) = 0. This space, with its , captures the topological constraints on the solutions, such as connectedness or if inherited from X and Y. Such s are fundamental in fiber bundle theory and , where they model restrictions over base spaces.

Colimits and quotients

In the category of topological spaces and continuous maps, colimits are constructed by first taking the colimit of the underlying diagram in the category Set and then endowing the resulting set with the with respect to the canonical cocone maps; this is the finest topology making those maps continuous and ensures the universal property holds in . This construction applies generally to small-indexed diagrams and is preserved by the from to Set, which has both left and right adjoints. The coproduct of a family of topological spaces \{X_i\}_{i \in I} is their disjoint union as sets, equipped with the coproduct topology in which a subset U is open if and only if U \cap X_i is open in X_i for every i \in I. The canonical inclusions i_i: X_i \to \coprod_{i \in I} X_i are continuous (in fact, open embeddings), and for any family of continuous maps f_i: X_i \to Z, there exists a unique continuous map \coprod f_i: \coprod X_i \to Z such that the diagrams commute; this characterizes the coproduct up to unique isomorphism in Top. Finite coproducts coincide with the standard disjoint union topology, and for a singleton family, the coproduct recovers the original space. Coequalizers in exist for any parallel pair of continuous maps f, g: X \to Y: the coequalizer is the set-theoretic quotient Y / \sim, where \sim is the smallest equivalence relation on the underlying set of Y containing pairs (f(x), g(x)) for all x \in X, equipped with the quotient topology making the canonical projection q: Y \to Y / \sim continuous and open. A subset V \subseteq Y / \sim is open if and only if q^{-1}(V) is open in Y, and for any continuous h: Y \to Z with h \circ f = h \circ g, there is a unique continuous \overline{h}: Y / \sim \to Z such that \overline{h} \circ q = h, satisfying the coequalizer universal property. This topology is the final one with respect to q, ensuring the construction is categorical. Pushouts, as binary colimits, arise from spans X \leftarrow A \to Y of continuous maps f: A \to X and g: A \to Y; the pushout is the of the X \sqcup Y by the identifying f(a) with g(a) for all a \in A, endowed with the topology from the induced map on the . This yields the universal property: for any Z with continuous u: X \to Z and v: Y \to Z such that u \circ f = v \circ g, there is a unique continuous w: X \cup_A Y \to Z making the triangles commute. When f is an of a A \subseteq X, the pushout is the attaching space obtained by gluing Y to X along A via g, a fundamental construction in . Representative examples illustrate these colimits. For a of a Y into open subsets \{U_i\}, the Y / \sim collapsing each U_i to a point inherits the quotient topology, often yielding a if the U_i cover Y. In the pushout construction, attaching the n-disk D^n to a base X along its S^{n-1} via a continuous \phi: S^{n-1} \to X produces the quotient X \cup_\phi D^n, where points on the boundary are identified according to \phi; for X = S^{n-1} and \phi the , attaching two disks yields the n- S^n with its standard . Such attaching spaces form the building blocks of CW-complexes, where colimits iteratively construct more complex topological from simpler cells.

Advanced properties

Monomorphisms and epimorphisms

In the category Top of topological spaces and continuous maps, a monomorphism is defined as a f: X \to Y that is left-cancellative: whenever continuous maps g, h: Z \to X satisfy f \circ g = f \circ h, it follows that g = h. These monomorphisms are precisely the injective continuous functions, as the injectivity ensures that distinct points in the domain map to distinct points, preserving the cancellation property under composition with continuous maps. Such maps are also known as topological embeddings, since they equip the image f(X) with the from Y, making f a onto its image. An in is a f: X \to Y that is right-cancellative: whenever continuous maps g, h: Y \to Z satisfy g \circ f = h \circ f, it follows that g = h. These epimorphisms are characterized as the continuous maps whose images are dense in the codomain, meaning the closure of f(X) equals Y. Continuous surjections are always epimorphisms, since a surjective image is necessarily dense, but the converse does not hold: epimorphisms need not be surjective. This contrasts with the category Set of sets and functions, where monomorphisms are exactly the injective functions and epimorphisms are exactly the surjective functions. In Top, the topological structure allows non-surjective epimorphisms, such as the inclusion of a proper dense subspace; for instance, the inclusion map i: \mathbb{Q} \to \mathbb{R} (with the standard topologies) has dense image but is not surjective, yet it is an epimorphism because any two continuous functions from \mathbb{R} to another topological space that agree on \mathbb{Q} must agree everywhere by density and continuity. Not all epimorphisms in are quotient maps, which are the regular epimorphisms (those expressible as coequalizers) and require both surjectivity and the quotient topology on the codomain. The inclusion i: \mathbb{Q} \to \mathbb{R} exemplifies a non-regular epimorphism, as its image is not closed and does not induce a quotient structure.

Subobject classifier absence

In the category of topological spaces and continuous maps, a subobject of an object X is an equivalence class of monomorphisms into X, where two monomorphisms m: Y \to X and m': Y' \to X are equivalent if there exists an isomorphism i: Y \to Y' such that m' \circ i = m. Regular subobjects are represented by closed embeddings, since the regular monomorphisms in —which pull back to form new regular monomorphisms—are precisely the embeddings of closed subspaces with the subspace topology. However, not all monomorphisms are regular; for example, the inclusion of an open subspace is a monomorphism but not regular. The category Top lacks a subobject classifier, which would be an object \Omega equipped with a global truth morphism \tau: 1 \to \Omega (where $1 is the terminal object, a singleton space) such that every subobject S \hookrightarrow X corresponds uniquely to a characteristic morphism \chi: X \to \Omega via the pullback square defining S as the equalizer of \chi and \tau. This absence arises because not all monomorphisms in Top are regular (i.e., equalizers), so no single \Omega can classify all subobjects via regular embeddings; the diverse topologies and non-regular monomorphisms prevent uniform classification of varying subspace structures. Consequently, Top is not an elementary topos, as the existence of a subobject classifier (together with finite limits, which Top does possess) would imply additional structure such as cartesian closedness. In contrast, the category Set of sets and functions is an elementary topos where the subobject classifier is the power-set object P(1), with characteristic functions classifying subsets. Related to this, Top lacks exponential objects in general: while topological function spaces Y^X can be constructed (e.g., via the compact-open topology when X is locally compact Hausdorff), they do not always represent the hom-sets \mathrm{Hom}(- \times X, Y) categorically, failing the universal property required for exponentials. This further underscores the absence of topos-like structure, as a subobject classifier would force the existence of such exponentials.

Relationships to other categories

Comparison with uniform spaces

The category Unif of spaces consists of sets equipped with a uniform structure, defined by a of entourages (symmetric subsets of the ) satisfying the uniform axioms: reflexivity, , and via entourages. These structures generalize spaces and topological groups, providing a framework for without relying on distances. Every uniform structure induces a unique on the underlying set, which is Hausdorff if the uniformity is separated and always completely regular. Morphisms in Unif are uniformly continuous functions, meaning that for every entourage V in the codomain's uniformity, there exists an entourage U in the domain's uniformity such that f \times f (U) \subseteq V. is stricter than mere with respect to the induced topologies: while every uniformly continuous map is continuous, the reverse fails in general. For instance, on the real line with the standard uniformity, the function f(x) = x^2 is continuous but not . The forgetful functor U: \mathbf{Unif} \to \mathbf{Top} maps each uniform space to its underlying topological space with the induced topology and sends uniformly continuous maps to continuous maps. This functor is faithful, as it preserves the underlying functions exactly, but not full, since continuous maps between induced topologies need not be uniformly continuous relative to the given uniform structures. Moreover, U is not essentially surjective, as only completely regular topological spaces arise as induced topologies from uniform structures; non-completely regular spaces, such as the Sierpiński space \{0,1\} with open sets \{\emptyset, \{0\}, \{0,1\}\}, admit no compatible uniform structure, since no continuous function to [0,1] separates the closed point \{1\} from $0$. Although U has a left adjoint \mathbf{Cont}: \mathbf{Top} \to \mathbf{Unif}, which first coreflects a topological space to its completely regular modification and then endows it with the fine uniformity (the largest uniformity inducing the topology), this adjunction does not yield an . The unit of the adjunction is an precisely on completely regular objects, but non-completely regular topologies like the map to a different (regularized) space, highlighting that Top encompasses structures incompatible with uniformity. Key differences between Top and Unif also manifest in their constructions. While Top permits arbitrary topologies, including indiscrete ones on sets with more than one point (which are vacuously completely regular but lack non-trivial uniform properties beyond the trivial entourage X \times X), Unif enforces uniformity axioms that ensure concepts like Cauchy filters are well-defined globally, not just locally as in topologies. Colimits in Unif, such as quotients, preserve the uniform structure via the quotient uniformity (the finest uniformity making the projection uniformly continuous), ensuring the result remains a ; in contrast, quotients in Top may yield non-completely regular topologies, exiting the image of U. For example, quotienting a uniformizable space by an in Top can produce a non-uniformizable quotient, whereas the corresponding colimit in Unif retains compatibility.

Functors to and from Top

The singular functor, often denoted Sing or S, assigns to each topological space X the simplicial set whose n-simplices are the continuous maps from the standard n-simplex \Delta^n to X. This functor Sing: Top \to sSet maps continuous functions between spaces to simplicial maps and forms a right adjoint to the geometric realization functor |\cdot|: sSet \to Top, enabling the transfer of homotopy-theoretic tools from simplicial sets to topological spaces. Not every topological space admits a compatible , so there is no from the full Top to the Met of spaces that preserves all structure. However, the full subcategory of metrizable spaces in Top is equivalent to Met via the that equips a metrizable space with one of its , though this equivalence is not canonical since are not unique; continuous maps between metrizable spaces correspond to non-expansive maps only if the is fixed appropriately, rendering the not full in general. The category Loc of locales, defined as the opposite of the category Frm of frames (complete Heyting algebras), relates to Top via the functor that sends a topological space X to the frame of its open sets \mathcal{O}(X), which is left adjoint to the sobrification functor; sobrification assigns to a frame its associated sober space, a reflective subcategory of Top where points correspond bijectively to prime ideals, thus providing a "pointless" generalization of topology that rectifies non-sober spaces. The Top embeds into broader categorical frameworks such as the of sheaves on a , where a X corresponds to the sheaf of sections over the posetal of its open sets, but this embedding is not fully faithful in general since sheafification may identify distinct continuous maps or spaces under the associated . A fundamental example of an adjunction involving Top is the pair consisting of the discrete topology functor D: Set \to Top, which equips a set with the discrete topology (all subsets open), and the underlying set functor U: Top \to Set, which forgets the topology; D \dashv U since continuous maps from a discrete space to a are precisely the functions to its underlying set, illustrating Top's concrete structure over Set.