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Compactly generated space

In , a compactly generated space (also known as a k-space) is a X in which every subset A ⊆ X is closed A ∩ K is closed in K for every compact subset K ⊆ X. This condition ensures that the topology on X is the coarsest one coherent with its compact subspaces, meaning it is generated by the compact subsets in a precise sense. The concept was introduced by Norman Steenrod in 1967 as part of constructing a "convenient category" of topological spaces suitable for and , where the usual fails to be Cartesian closed due to poorly behaved function spaces. In this category, often denoted or restricted to compactly generated weak Hausdorff spaces (CGWH) to ensure better separation properties, infinite products, coproducts, quotients, and mapping spaces preserve key topological features, making it ideal for studying limits and colimits in contexts. Michael C. McCord further developed the framework in 1969, emphasizing weak Hausdorff k-spaces and their role in infinite-dimensional topology. Compactly generated spaces exhibit several important properties: they are closed under the formation of closed subspaces, arbitrary coproducts, and quotients; the category is Cartesian closed, with the on function spaces yielding compactly generated results under mild conditions. Notable examples include all metric spaces, first-countable Hausdorff spaces, locally compact Hausdorff spaces, CW-complexes, and smooth manifolds, which collectively cover most spaces encountered in classical topology and geometry. These spaces are particularly valuable in , where they facilitate the study of , infinite symmetric products, and classifying spaces without pathological counterexamples.

Definitions

Core Definition

A X is called compactly generated, or equivalently a k-space, if every A \subseteq X is closed in X A \cap K is closed in the on K for every compact K \subseteq X. This condition ensures that the of X is fully determined by its behavior on compact subsets, providing a practical framework for verifying closedness without examining the entire . Subsets A \subseteq X satisfying the right-hand side of the defining condition—namely, that A \cap K is relatively closed in every compact K \subseteq X—are termed k-closed sets. The collection of all k-closed sets forms a on the underlying set of X (via complements being k-open), called the k-topology or compactly generated topology on X, which coincides with the original topology precisely when X is compactly generated. This k-topology is the finest topology on X such that the induced on every compact subset K \subseteq X remains unchanged from the original; in other words, it preserves the intrinsic topologies of all compact subsets while ensuring no coarser topology would suffice for this purpose. In this framework, compact subsets play a generating role by dictating the global structure of the space: a map f: X \to Y into another space Y is continuous if and only if its restriction f|_K: K \to Y is continuous for every compact K \subseteq X. This property underscores the utility of compactly generated spaces in contexts where compacta suffice to capture essential topological features, assuming familiarity with basic notions such as topological spaces and compactness.

Equivalent Formulations

A compactly generated space, also known as a , admits several equivalent formulations that highlight its topological structure in relation to compact subsets. One such characterization is that the on the space X is coherent with respect to its family of compact subsets, meaning that a U \subseteq X is open U \cap K is open in the on K for every compact K \subseteq X. This formulation emphasizes how the overall is precisely determined by the topologies on compact subsets, ensuring that openness is detectable locally on compacts. An alternative perspective defines compactly closed sets within X: a subset A \subseteq X is compactly closed if A \cap K is closed in K for every compact K \subseteq X. The space X is then compactly generated if every compactly closed set is closed in X. This condition captures the idea that closed sets are those whose "restrictions" to compact subsets remain closed, providing a closure operator that aligns with the compactly generated topology. Another equivalent formulation arises from the k-ification process, which equips X with the k-topology, the finest on the underlying set of X such that the on every compact subset agrees with the original; this is the with respect to the family of all maps from compact Hausdorff spaces into X. In categorical terms, this corresponds to the coreflection of X in the category of k-spaces, yielding a topology where the behaves well under limits and colimits involving compacts. To see the equivalence between the core definition—where closed sets are intersections of closures containing compacts—and the coherent topology formulation, note that in a compactly generated space, compact subsets "detect" openness: if a set U is open, then U \cap K is open in every compact K, and conversely, the saturation of U under the k-closure operator (generated by compact intersections) yields the full . This equivalence holds because the k-ification preserves the property that maps from compacts are continuous, and the coherent condition ensures no finer is needed, as verified by showing that any non-open set fails detection on some compact.

Role of Compact Subsets

In compactly generated spaces, also known as k-spaces, the is generated by a specified family of subsets through the construction of the k-topology, which is the finest topology on the underlying set such that each of these compact subsets remains compact and inherits its original . This process ensures that the open sets in the k-topology are precisely those subsets whose intersections with every compact subset from the family are open in the respective subspace topologies. Such a generation mechanism highlights the foundational role of compact subsets, as they dictate the coarsest possible refinement of the while preserving their compactness, thereby avoiding pathologies where additional structure would be required to maintain these properties. A key detection property of compact subsets in k-spaces is their ability to determine the of into the . Specifically, for a f: Y \to X where X is a , f is if and only if the restriction f|_C: C \to X is for every compact subset C \subseteq Y. Equivalently, holds if and only if the f \circ g: K \to X is for every compact K and every g: K \to Y. This property underscores how compact subsets act as "probes" that fully capture the topological structure of X, allowing global to be verified locally on these subsets. In k-spaces, compact subsets suffice to capture openness because the topology is defined such that a is open precisely when its preimage under continuous maps from compact Hausdorff spaces is open, eliminating the need for finer topologies that might arise in non-k-spaces where compact subsets fail to detect all open sets. This sufficiency arises from the universal property of the k-ification, which quotients the space to enforce compactness preservation, ensuring that the resulting is both with respect to maps from compact spaces and fully determined by the on those subsets. Consequently, compact subsets provide a complete and efficient basis for topological verification in these spaces.

Historical and Motivational Context

Origins in

In the mid-20th century, faced challenges in extending to infinite constructions within the standard , where infinite products and coproducts often exhibited pathologies that disrupted the expected behavior of homotopy groups and related invariants. For example, the on an infinite product of spaces might fail to preserve the type necessary for consistent computations of homotopy colimits or limits, complicating the analysis of weak equivalences and theories. This issue was particularly acute for spaces like infinite CW-complexes, where the natural colimit topology—generated by compact subcomplexes—was needed to ensure that global properties are controlled by compact subsets, thereby avoiding inconsistencies in infinite limits and preserving invariants such as higher groups. The development of compactly generated spaces arose directly from these needs, providing a refinement of the that better supports the foundational tools of , including and the study of function spaces under the . By mid-century, as and theories gained prominence, topologists recognized that the naive lacked sufficient structure for handling arbitrary colimits while maintaining homotopy-theoretic coherence, especially in contexts involving non-locally compact domains. Compactly generated topologies addressed this by ensuring that continuous maps and homotopies are determined by their restrictions to compact subsets, aligning infinite constructions with finite approximations and enabling reliable extensions of finite-dimensional results to infinite cases. These motivations trace back to informal developments in the late 1940s, notably lectures by in 1948–1949, which introduced the notion of k-spaces to resolve issues in function space compactness within homotopy contexts. The concept first appeared in print in a 1950 paper by David Gale, who applied k-spaces to extend the Arzelà–Ascoli theorem to broader classes of s, demonstrating their role in controlling and for applications in . This work underscored the practical necessity of such spaces for preserving invariants under infinite operations, setting the stage for later refinements in the field.

Steenrod's Framework and Improvements

In 1967, Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces as a more suitable framework than the category of all topological spaces () for studying infinite products and coproducts in . Steenrod argued that in CGH, these constructions behave better, preserving key topological properties essential for , such as the continuity of function spaces. A central aspect of Steenrod's framework was the imposition of the Hausdorff condition, which ensures the uniqueness of limits and colimits, thereby providing a solid foundation for homotopy-theoretic constructions that were problematic in the broader category . This refinement addressed issues arising from non-Hausdorff spaces, where infinite products might fail to reflect the expected topological structure, and it facilitated the study of mapping spaces in a categorical setting. Subsequent developments simplified and generalized Steenrod's approach; in particular, Michael C. McCord in 1969 proposed the category of k-spaces, which are compactly generated without the strict Hausdorff requirement, allowing for broader applicability while retaining many desirable categorical properties. Further enhancements included the incorporation of weakly Hausdorff variants, where a space is weakly Hausdorff if the image of any continuous map from a compact Hausdorff space is closed, forming the category CGWH that balances generality with control over pathological behaviors. These advancements enabled a rigorous treatment of homotopy colimits and the geometric realization of singular complexes, resolving foundational challenges in by ensuring that colimits of diagrams over compactly generated spaces align with homotopy-theoretic expectations.

Illustrative Examples

Positive Examples

First-countable Hausdorff spaces are compactly generated, as the countable local basis at each point ensures that compact subsets suffice to determine the , with openness detected via intersections with these compacts. Metric spaces, such as spaces \mathbb{R}^n, exemplify this , where the metric induces a first-countable Hausdorff that aligns with compact generation. Locally compact Hausdorff spaces are also compactly generated, since in such spaces, compact subsets are closed and their intersections generate the full . Smooth manifolds and spaces further illustrate this, as their local compactness—arising from compact coordinate charts or bounded neighborhoods—ensures the topology is determined by compacta. CW-complexes, both finite and infinite, qualify as compactly generated spaces, with the skeleta serving as compact generators that dictate openness through their attachments. This property holds because any compact subset of a CW-complex is contained within a finite subcomplex, allowing the topology to be recovered from these compact pieces. Compact Hausdorff spaces are trivially compactly generated, as the entire space acts as a single compact subset from which the topology is immediately determined. Examples include the unit interval [0,1] or the 2-sphere S^2, where the compactness and Hausdorff separation jointly ensure the k-space condition.

Non-Examples and Pathologies

Another standard non-example is the square of the one-point compactification \alpha \mathbb{Q} of the rational numbers \mathbb{Q} equipped with the subspace topology from \mathbb{R}. Here, \alpha \mathbb{Q} = \mathbb{Q} \cup \{\infty\} with open sets being the open subsets of \mathbb{Q} and complements in \alpha \mathbb{Q} of compact subsets of \mathbb{Q}. While \alpha \mathbb{Q} itself is compactly generated (as a compact Hausdorff space), the product \alpha \mathbb{Q} \times \alpha \mathbb{Q} is not, since there exist subsets—such as certain "diagonals" involving the infinity points—that are closed relative to every compact subset but fail to be closed in the entire space, violating the k-condition. Infinite products can also fail the compactly generated property. For instance, the uncountable product \mathbb{R}^J where J is an uncountable index set is not compactly generated, even though each factor \mathbb{R} is. In this space, compact subsets are contained in countable subproducts (by properties of products of metric spaces), so they fail to "detect" the full topology; specifically, there are closed sets whose intersections with all compact subsets are closed, but the sets themselves are not closed overall. A key pathology arising in non-k-spaces is the existence of compactly closed sets that are not actually closed. Such sets intersect every compact subset in a closed set but remain non-closed globally, which disrupts the behavior of function spaces under the . In , this leads to issues where continuous maps and homotopies between non-k-spaces may not behave well under or , necessitating the k-ification process to restore desirable categorical properties like cartesian closedness.

Fundamental Properties

Behavior Under Set Operations

Closed subspaces of a k-space are themselves k-spaces, as the compact subsets of the subspace inherit the relative topology from compact subsets of the ambient space, preserving the compactly generated property. Arbitrary subspaces of a k-space, however, are not necessarily k-spaces; for instance, an open subspace may fail to have its topology generated by its compact subsets unless the subspace is locally compact or the original space satisfies additional conditions like local compactness. In such cases, the subspace topology aligns with the k-topology induced by its own compact subsets only when these coherency requirements hold. Quotients of k-spaces preserve the k-property under specific conditions on the . If the equivalence relation is closed in the product space, the resulting space is a k-space, since the identification map ensures that closed sets in the correspond to saturated closed sets in the whose intersections with compact subsets remain closed. More precisely, for an equivalence relation that is k-closed (meaning its intersection with compact subsets of the product is closed), the inherits the compactly generated topology. General quotients, however, may not be k-spaces, as arbitrary identifications can introduce sets whose closedness is not determined by compact subsets. A fundamental aspect of k-spaces is the behavior of closures under the compactly generated topology. A subset F \subseteq X is closed in the k-space X F \cap K is closed in every compact subset K \subseteq X. This characterization implies that the closure of any subset A \subseteq X is given by \cl(A) = \bigcup \{ \cl_K(A) \mid K \subseteq X \text{ compact} \}, where \cl_K(A) denotes the closure of A in the on K. This formula reflects how the k-topology ensures that limits and closures are controlled locally within compact subsets, providing a practical way to compute closures by restricting to compacts.

Continuity and Function Spaces

In compactly generated spaces, also referred to as , the continuity of maps is characterized by their restrictions to compact subsets. Specifically, if X is a and Y is any , then a f: X \to Y is the restriction f|_K: K \to Y is continuous for every compact subset K \subseteq X, where K inherits the from X. This compact-detection property arises because the on a is the finest topology making all maps from compact Hausdorff spaces into X continuous, ensuring that closed sets (and thus ) are determined locally on compacts. The space of continuous maps \operatorname{Map}(X, Y) from a k-space X to a topological space Y is endowed with the , generated as a subbasis by sets of the form \{f \in \operatorname{Map}(X, Y) \mid f(K) \subseteq U\}, where K \subseteq X is compact and U \subseteq Y is open. When X is a , \operatorname{Map}(X, Y) equipped with this topology inherits the k-space property, meaning its closed subsets are precisely those that are closed when intersected with images of compact subsets under evaluation maps. Moreover, for k-spaces X and Y, the on \operatorname{Map}(X, Y) coincides with the k-topology induced by the family of compact subsets of X, ensuring that the function space behaves coherently within the category of k-spaces. This structure simplifies continuity verification in , particularly for constructions involving . In , a singular n-simplex is a continuous map \sigma: \Delta^n \to X, where \Delta^n is the standard compact ; since \Delta^n is compact, the k-space property of X ensures that such maps are continuous precisely when they satisfy the topological conditions on their compact domain, facilitating computations without needing to check global preimages of closed sets.

Preservation in Constructions

The k-property of compactly generated spaces is preserved under various categorical constructions, ensuring that the category behaves well for and . In particular, products and coproducts maintain the compactly generated structure when appropriately equipped. The product X \times Y of two k-spaces, equipped with the , is a k-space. This extends to infinite products: the infinite \prod_{i \in I} X_i of k-spaces, initially taken in the product topology (Tychonoff topology), is k-ified to yield a k-space, often called the k-product topology. In the of compactly generated Hausdorff (CGH) spaces, this k-product avoids such as non-Hausdorff behavior in infinite products of Hausdorff spaces, preserving both the k-property and Hausdorff separation. For example, the infinite product of copies of the real line \mathbb{R} in the CGH category remains Hausdorff and compactly generated under this construction. In the category of k-spaces, products are defined as the k-ification of the ordinary , which for finite products coincides with the itself. Coproducts, realized as , always preserve the k-property: the disjoint union \coprod_{i \in I} X_i of any family of k-spaces is itself a k-space, without needing further k-ification, as the inherits the compactly generated structure directly from the components. This holds in both the general category and the CGH subcategory, making coproducts particularly straightforward for building larger spaces from compactly generated building blocks. Regarding limits and initial topologies, the k-property is preserved under pullbacks and other small limits when computed in the of topological spaces and then k-ified. Specifically, pullbacks along closed inclusions maintain the compactly generated nature, ensuring that the resulting space remains a k-space if the input spaces are. In the CGH setting, this preservation extends to equalizers and other limits, with the to sets reflecting these structures faithfully. Such constructions are crucial for forming spaces and other functional limits while retaining the compactly generated .

Categorical Structure

The Category of k-Spaces

The category \mathbf{K} of k-spaces has as objects the compactly generated topological spaces and as morphisms the continuous functions between them. It is a full subcategory of the category \mathbf{Top} of all topological spaces, but with the understanding that non-k-spaces are equipped with their k-ification to ensure the topology is coherent with compact subsets. This adjustment allows \mathbf{K} to include all objects of \mathbf{Top} densely, as every topological space admits a k-ification that preserves relevant topological invariants. A key subcategory is \mathbf{CGWH}, consisting of compactly generated weak Hausdorff spaces, which is closed under the formation of arbitrary limits and colimits. In \mathbf{CGWH}, products and coproducts inherit the compactly generated weak Hausdorff structure naturally, making it suitable for categorical constructions in . The inclusion functor from \mathbf{CGWH} to \mathbf{K} is both full and faithful, preserving the weak Hausdorff property. The category \mathbf{K} satisfies the axioms of a cartesian closed category, where the internal hom-object Y^X is equipped with the k-topology induced by the compact-open topology on mapping spaces. This closure property ensures that function spaces behave well under composition and evaluation, with the exponential law Z^{X \times Y} \cong (Z^Y)^X holding as a homeomorphism in \mathbf{K}. Moreover, \mathbf{K} supports a robust homotopy theory, where homotopy pullbacks coincide with ordinary pullbacks and weak homotopy equivalences are preserved under limits, facilitating computations in algebraic topology. The embedding of \mathbf{[Top](/page/Top)} into \mathbf{[K](/page/K)} via k-ification is dense, meaning every object in \mathbf{[K](/page/K)} is a colimit of objects from the image of \mathbf{[Top](/page/Top)}, but it is neither full nor faithful. Specifically, certain maps discontinuous in \mathbf{[Top](/page/Top)} become continuous after applying the k-ification to domain and , reflecting the finer imposed on non-k-spaces. This reflective adjunction, with k-ification as the right to the , ensures that \mathbf{[K](/page/K)} captures essential categorical structure while mitigating pathologies in \mathbf{[Top](/page/Top)}.

K-Iification Process

The k-ification process constructs the compactly generated topology on an arbitrary X, denoted kX, by refining the original to ensure it is determined solely by its compact subsets. Specifically, a U \subseteq X is declared open in kX if and only if U \cap K is open in the on K for every compact K of the original X. This yields the finest on the underlying set of X such that each compact K of X remains compact in kX and retains its original . A subbasis for the topology on kX consists of all sets of the form V \cap K, where V is open in the original on X and K is compact in X. The identity map \mathrm{id}_X: X \to kX is continuous, and for any compactly generated space Y, a map f: X \to Y is continuous if and only if its with \mathrm{id}_X is continuous. This construction defines a k: \mathbf{[Top](/page/Top)} \to \mathbf{K} from the category of topological spaces to the category of compactly generated spaces, which is right to the functor \mathbf{K} \hookrightarrow \mathbf{[Top](/page/Top)}. The adjunction implies that continuous maps into kX from compactly generated spaces correspond bijectively to continuous maps into X. The functor k is idempotent, meaning k(kX) \cong kX for any space X, and if X is already compactly generated, then kX is homeomorphic to X via the identity map. This idempotence ensures that the subcategory of compactly generated spaces is reflective in \mathbf{Top}. In applications, k-ification addresses pathologies in non-compactly generated spaces, particularly in algebraic topology; for instance, the geometric realization of a simplicial set is typically equipped with the k-ified topology to guarantee it is a compactly generated space, facilitating homotopy-theoretic computations such as colimits and mapping spaces.

Connections to Other Topological Categories

Relation to Hausdorff and Locally Compact Spaces

Every locally compact is a , as compact subsets are closed in such spaces and generate the topology in the required manner. This inclusion follows from the fact that in a locally compact , the closure of any set intersects compact sets appropriately to ensure the property. The subcategory of compactly generated Hausdorff (CGH) spaces imposes the Hausdorff separation axiom on k-spaces, making it a convenient category for and . In contrast, general k-spaces need not be Hausdorff, allowing for non-separated topologies that remain useful in certain contexts where separation is not essential. Although every locally compact is a k-space, the converse does not hold; for instance, the space of rational numbers \mathbb{Q} endowed with the from \mathbb{R} is a k-space, being first countable, but it is not locally compact at any point. k-spaces thus encompass a wider class than locally compact spaces, accommodating examples without local compactness. Both k-spaces and locally compact play key roles in manifold theory and the study of Lie groups, where typical examples like finite-dimensional manifolds are locally compact and hence k-spaces. However, the broader scope of k-spaces proves advantageous in infinite-dimensional settings, such as infinite CW-complexes in , where local compactness fails but the compactly generated condition preserves essential categorical properties.

Weak Hausdorff Variant

A topological space X is defined to be weak Hausdorff if, for every compact Hausdorff space K and every continuous map f: K \to X, the image f(K) is closed in X. This condition is weaker than the standard Hausdorff separation axiom, as it does not require disjoint open neighborhoods for distinct points, but it ensures that compactly generated structures behave well under continuous images from compact Hausdorff domains. Compactly generated weak Hausdorff spaces, often abbreviated as CGWH spaces, combine the compactly generated () condition with weak Hausdorff separation. These spaces form a subcategory of topological spaces that is particularly suitable for , where the classical Quillen model structure on the restricts naturally to CGWH objects, with weak equivalences as the weak equivalences and Serre fibrations as fibrations. In this setting, cofibrations are closed inclusions of closed cell complexes, providing a structure that supports homotopy limits and colimits effectively. This refinement addresses limitations in Steenrod's original compactly generated Hausdorff (CGH) framework by relaxing the while preserving essential categorical properties. A key property of weak Hausdorff spaces is that every compact subset is closed. For k-spaces that are also weak Hausdorff, this implies that limits in the , such as products and equalizers, inherit the weak Hausdorff property, and the to topological spaces creates all limits, facilitating computations in categories. Moreover, the of CGWH spaces is cartesian closed, allowing function spaces to be well-behaved and enabling the internal hom to be realized as a topological space within the . The weak Hausdorff condition offers advantages in applications where full Hausdorff separation is too restrictive, such as in the study of algebraic varieties over non-algebraically closed fields equipped with the . In these cases, the space is generally not Hausdorff—for instance, any two nonempty open sets intersect—but compact subsets remain closed, preserving arguments essential for geometric and homological studies. This makes CGWH spaces a natural setting for bridging and , accommodating examples like the over the reals without sacrificing the closure of compact images.