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Open and closed maps

In topology, an open map is a function between topological spaces that maps open sets to open sets, while a closed map maps closed sets to closed sets; these properties highlight how certain functions preserve the openness or closedness of subsets without necessarily requiring continuity or bijectivity.^[1]^[2] Formally, given topological spaces X and Y, a map f: X \to Y is open if for every open subset U \subseteq X, the image f(U) is open in Y, and it is closed if for every closed subset C \subseteq X, the image f(C) is closed in Y.^[1]^[2] These definitions do not imply continuity, as there are no general relationships between openness, closedness, and continuity for arbitrary maps.^[2] A key application arises in the characterization of homeomorphisms: a continuous bijection f: X \to Y is a homeomorphism if and only if it is open (or equivalently, closed), ensuring it preserves all topological properties.^[1]^[2] Classic examples include the inclusion map (0,1) \hookrightarrow \mathbb{R}, which is open but not closed, since open sets in the subspace map to open sets in \mathbb{R}, but for example, the set \{1/n \mid n = 2,3,\dots \} is closed in (0,1) but its image is not closed in \mathbb{R}; conversely, the inclusion [0,1] \hookrightarrow \mathbb{R} is closed but not open.^[1] Projection maps, such as \pi_1: \mathbb{R} \times \mathbb{R} \to \mathbb{R} given by (x,y) \mapsto x, are open but not closed, as they map open sets like open balls to open intervals while sending some closed sets, like lines parallel to the y-axis, to non-closed images.^[3]^[2] In quotient topology, open or closed surjective continuous maps yield quotient maps, which define the finest topology on the codomain making the map continuous; however, not all quotient maps are open or closed.^[3]

Definitions

Open maps

In topology, an open map is a function between topological spaces that preserves the openness of sets under direct images. Specifically, a map f: X \to Y between topological spaces X and Y is defined to be open if, for every open set U \subseteq X, the image f(U) is an open set in Y.^[3] Formally, this condition can be stated as: for all U \in \tau_X, where \tau_X denotes the topology on X, it holds that f(U) \in \tau_Y, with \tau_Y the topology on Y.^[4] This definition appears in standard treatments of general topology and underscores the map's behavior with respect to the open sets of the codomain.^[5] The role of an open map lies in its preservation of structural openness from the domain to the codomain, ensuring that the topological features of open subsets in X translate directly into open subsets in Y.^[3] Unlike continuous maps, which focus on preimages of open sets, open maps emphasize the direct images f(U), thereby highlighting how the topology \tau_Y on Y interacts with the mapping process. This preservation is crucial in contexts where the openness in the codomain must align with the mapped structure from the domain, without requiring surjectivity or other additional properties.^[4]

Closed maps

A map f: X \to Y between topological spaces is called a closed map if the image under f of every closed subset of X is closed in Y.^[6] Formally, f is closed if for every closed set C \subseteq X, f(C) belongs to the family of closed sets in Y, denoted \tau_Y^c, which consists of the complements of open sets in the topology \tau_Y on Y.^[7] This property ensures that closed maps preserve the "closedness" of subsets under the function's action, distinguishing them from the dual concept of open maps, which instead map open sets to open sets.^[8] In particular, the focus on closed sets—defined as complements of open sets—avoids direct overlap with openness conditions, emphasizing instead the behavior of boundaries and limits in the codomain.^[9] Closed maps are useful for controlling the closure of images of subsets.^[6]

Characterizations and variants

Equivalent conditions

A map f: X \to Y between topological spaces is open if and only if f(A^\circ) \subseteq [f(A)]^\circ for every subset A \subseteq X, where \circ denotes the interior operator.^[10] Dually, a map f: X \to Y is closed if and only if \overline{f(A)} \subseteq f(\overline{A}) for every subset A \subseteq X, where the bar denotes the closure operator.^[10] This condition ensures that the image under f respects closures in a manner symmetric to how the interior condition characterizes openness. For surjective maps, additional equivalences arise in terms of preimages and saturation: a surjective f is open if the saturation of every open set in X (with respect to the fibers of f) is open.^[10] A bijective map f: X \to Y is a homeomorphism if and only if it is open; equivalently, if and only if it is closed.^[11] This holds because openness of a bijection implies bicontinuity, as the inverse maps open sets to open sets via the direct images under f.

Strong and relative versions

In topology, the usual notion of an open map requires that the image of every open set in the domain is open in the entire codomain. Specifically, for a function f: X \to Y between topological spaces X and Y, f is open if for every open set U \subseteq X, the set f(U) is open in Y.^[12] Analogously, a closed map is one where the image of every closed set in X is closed in Y.^[12] To address non-surjective maps, relative versions of these concepts are defined with respect to the subspace topology on the image f(X) \subseteq Y. A map f: X \to Y is a relative open map if for every open set U \subseteq X, the set f(U) is open in the subspace f(X), meaning there exists an open set V \subseteq Y such that f(U) = V \cap f(X).^[13] Similarly, f is a relative closed map if the image of every closed set in X is closed in the subspace f(X), i.e., there exists a closed set W \subseteq Y such that f(C) = W \cap f(X) for every closed C \subseteq X.^[13] These relative notions weaken the usual versions by focusing openness or closedness locally within the image rather than the full codomain. When f is surjective, so that f(X) = Y, the subspace topology on f(X) coincides with the topology on Y, and thus the usual and relative versions of openness (or closedness) are equivalent.^[14] In non-surjective cases, the relative condition is strictly weaker than the usual one; for instance, any inclusion map i: A \hookrightarrow Y of a subspace A \subseteq Y is relative open (and relative closed) onto its image by the definition of the subspace topology, but it is open only if A itself is open in Y.^[13] These variants emerged in the development of quotient topology, where the standard constructions assume surjective maps to ensure the quotient map is both continuous and open (or closed), but extensions to non-surjective settings, such as in general identification spaces or momentum maps, necessitated the relative notions to preserve topological properties within the image.^[15]

Examples

Continuous maps that are open or closed

A fundamental example of a continuous open map arises in product spaces. Consider the projection map \pi_X: X \times Y \to X defined by \pi_X(x, y) = x, where X and Y are topological spaces equipped with the product topology on X \times Y. This map is continuous by the definition of the product topology, as the preimage of any open set U \subseteq X is U \times Y, which is open in the product space.^[16] Moreover, \pi_X is open: the image of a basic open set U \times V \subseteq X \times Y (with U open in X and V open in Y) is U, which is open in X. This holds more generally for arbitrary open sets in the product, as they are unions of such basic opens, and images preserve unions. To verify explicitly, suppose W \subseteq X \times Y is open; then W = \bigcup_i (U_i \times V_i) for opens U_i \subseteq X and V_i \subseteq Y, so \pi_X(W) = \bigcup_i U_i, which is open in X. Thus, projections illustrate how continuity and openness combine in coordinate extractions from products.^[17] Another illustrative case is the inclusion map i: A \to X where A is a closed subset of a topological space X. The map i(a) = a is always continuous when A inherits the subspace topology from X. Furthermore, i is a closed map precisely when A is closed in X: for any closed set C \subseteq A, C is closed in X (as the intersection of a closed set in X with the closed A), and i(C) = C is thus closed in X. This property highlights how embeddings of closed subsets preserve closedness under continuous inclusion, preserving the topological structure in a restrictive yet faithful manner.^[18]

Maps that are open or closed but not continuous

A classic example of a map that is open but not continuous is the identity map on a set X with at least two elements, where the domain is equipped with the indiscrete topology (also known as the trivial topology, in which the only open sets are \emptyset and X) and the codomain is equipped with the discrete topology (in which every subset is open). The images of the open sets in the domain are \emptyset (which is open in the discrete topology) and X (which is open in the discrete topology), so the map is open.^[19] However, it is not continuous, because the preimage of a singleton \{x\} (which is open in the discrete codomain) is \{x\}, and singletons are not open in the indiscrete domain unless |X| = 1.^[19] This example also illustrates that the map is closed, since the closed sets in the indiscrete domain are \emptyset and X, whose images are closed in the discrete codomain.^[19] Another example of an open map that is not continuous arises in the cofinite topology on an uncountable set X, where the open sets are \emptyset and the complements of finite subsets of X. Consider the identity map \mathrm{id}: (X, \tau_\mathrm{cofinite}) \to (X, \tau_\mathrm{discrete}), where the codomain has the discrete topology. The image of any nonempty open set in the domain is cofinite in X, and every subset (including cofinite sets) is open in the discrete codomain, so the map is open. It is not continuous, however, because the preimage of a singleton \{x\} (open in the discrete codomain) is \{x\}, which has uncountable complement and thus is not open in the cofinite topology. For a closed map that is not continuous, consider the identity map \mathrm{id}: (\mathbb{R}, \tau_\mathrm{cofinite}) \to (\mathbb{R}, \tau_\mathrm{standard}), where the domain has the cofinite topology and the codomain has the standard Euclidean topology. The closed sets in the cofinite topology are the finite subsets and \mathbb{R} itself; their images under the identity are finite (hence closed in the standard topology) or \mathbb{R} (closed), so the map is closed.^[20] The map is not continuous, as the preimage of the open interval (0,1) (open in the standard codomain) is (0,1), whose complement in \mathbb{R} is (-\infty,0] \cup [1,\infty) (uncountable, hence not finite), so (0,1) is not open in the cofinite domain.^[20] A further illustration of a closed but discontinuous map is found in the countable Fort space, a topology on a countable infinite set X = \{p\} \cup Y where Y is countably infinite and p \notin Y: the open sets are all subsets of Y and all cofinite subsets of X.^[21] The projection map \pi: X \to Y (sending p to some fixed point in Y and fixing points in Y) to Y equipped with the discrete topology is closed, as the images of closed sets (which include the finite subsets of Y and all subsets containing p) are finite or all of Y, both closed in the discrete topology. It is not continuous, because the preimage of a singleton \{y\} \subset Y (open in the discrete codomain) is \{y\} if y \neq \pi(p), which does not contain p and is not all of X, hence not open in the Fort space topology.

Sufficient conditions

Conditions involving continuity

In topology, a fundamental result concerning closed maps involves continuous bijections between specific types of spaces. Specifically, if X is a compact space and Y is a Hausdorff space, then any continuous bijection f: X \to Y is a closed map.^[22] Moreover, such a bijection is in fact a homeomorphism, as its inverse is also continuous. This theorem highlights how continuity, combined with compactness of the domain and Hausdorff separation in the codomain, suffices to ensure the map is closed and thus a topological embedding in the bijective case. Another key condition arises in the context of quotient maps, which are surjective continuous maps that identify the topology of the codomain via saturation of preimages. A continuous surjective map f: X \to Y that is also open is necessarily a quotient map.^[3] This follows because the openness ensures that the preimage condition for open sets in Y aligns precisely with the quotient topology definition. Such maps are particularly useful in constructing new spaces by gluing or identification while preserving topological structure. For injective continuous maps, openness provides a sufficient condition for the map to be an embedding, meaning it is a homeomorphism onto its image equipped with the subspace topology. If f: X \to Y is a continuous injection and open (onto its image), then f embeds X as an open subspace of Y.^[23] This property is essential in differential topology and manifold theory, where embeddings distinguish submanifolds from mere immersions. Continuous maps inherently preserve connectedness: if X is a connected space and f: X \to Y is continuous, then f(X) is connected in Y.^[24] When the continuous map is additionally open, this preservation extends to ensuring the image is an open connected subset, reinforcing the map's role in maintaining global connectivity properties under openness.^[24]

Conditions from topological properties

A fundamental sufficient condition for a continuous map f: X \to Y to be closed arises from compactness in the domain and the Hausdorff property in the codomain. Specifically, if X is compact and Y is Hausdorff, then f maps closed subsets of X to closed subsets of Y. This follows because the continuous image of a closed subset of X—which is compact—is closed in the Hausdorff space Y.^[25] Local homeomorphisms provide a condition for openness independent of global compactness. A map f: X \to Y is a local homeomorphism if every point in X has an open neighborhood on which f restricts to a homeomorphism onto its image, which is open in Y. Consequently, f is an open map, as the image of any open set in X is a union of such open images. Covering maps, being surjective local homeomorphisms between topological spaces (typically with path-connected and locally path-connected base), inherit this openness property.^[26] Proper maps offer another intrinsic condition for closedness, particularly in settings involving locally compact spaces. A continuous map f: X \to Y is proper if the preimage of every compact subset of Y is compact in X. When Y is Hausdorff, such a proper map is closed, since the image of a closed subset of X has compact preimages under the inverse, ensuring closure in Y. This generalizes the compact-to-Hausdorff case, where compactness makes every continuous map proper.^[27] In the context of linearly ordered topological spaces (LOTS), known as chains with the order topology, strictly monotone maps—strictly order-preserving functions—satisfy closedness under surjectivity. A surjective strictly monotone map f: X \to Y between chains maps closed sets to closed sets, as the order preservation ensures that intervals and their complements behave accordingly under the map. In fact, such surjections are homeomorphisms, reinforcing their closed (and open) nature.^[28]

Properties and relations

Preservation under composition

In topology, the composition of two open maps is open. If f: X \to Y and g: Y \to Z are open maps between topological spaces, then for any open set U \subseteq X, f(U) is open in Y, and thus g(f(U)) is open in Z.^[29] Similarly, the composition of two closed maps is closed: if C \subseteq X is closed, then f(C) is closed in Y, and g(f(C)) is closed in Z.^[30] However, the composition of an open map with a closed map (in either order) need not be open or closed.^[31] When the maps are surjective, the preservation of openness or closedness under composition holds without additional conditions, as the property relies only on the images of open or closed sets, and surjectivity ensures the images cover the codomain but is not required for the basic result. At a higher level, the classes of open and closed maps are closed under composition, enabling the definition of categories where the objects are topological spaces and the morphisms are open maps (or closed maps). Such categories have been studied, though they lack certain limits, such as binary products in the case of open maps.^[32] This structure relates to monoidal or enriched category theory in topology, where open maps play a role in models of spatial processes.

Connections to quotient maps and homeomorphisms

A continuous surjective map p: X \to Y between topological spaces is a quotient map if it endows Y with the quotient topology, meaning a subset U \subseteq Y is open precisely when p^{-1}(U) is open in X. Since p is continuous and surjective, the forward direction holds automatically, but the converse requires additional structure. If p is also an open map, then for any U \subseteq Y with p^{-1}(U) open, set V = p^{-1}(U), which is open, and U = p(V); openness of p ensures U is open in Y. Similarly, if p is closed, the argument uses closed complements: if p^{-1}(U) is open, its complement is closed, and surjectivity with closedness of p implies the complement of U is closed, so U is open.^[33] Homeomorphisms, which are topological isomorphisms, can be characterized using openness or closedness for bijective continuous maps. Specifically, a bijective continuous map f: X \to Y is a homeomorphism if and only if it is open, meaning it maps open sets to open sets; equivalently, if it is closed. This follows because for a bijection, the inverse f^{-1} is continuous if and only if f is open: the preimage under f^{-1} of an open set in X is the image under f of that set in Y, so openness of f ensures continuity of f^{-1}. The closed characterization is analogous, using closed sets. In algebraic topology, covering maps provide a key connection, as the universal covering space of a path-connected, locally path-connected space is a covering map, which is a local homeomorphism and hence an open map. A local homeomorphism p: \tilde{X} \to X means every point in \tilde{X} has a neighborhood homeomorphic via p to a neighborhood in X, implying openness since local images of open sets are open. Universal covers, being simply connected covering spaces, thus identify discrete fibers evenly while preserving openness, facilitating computations of fundamental groups via the Galois correspondence between subgroups and covers.^[34] Open quotient maps play a crucial role in algebraic topology for constructing spaces by identifying fibers, such as in group actions where orbits form fibers. When a group G acts continuously on a space X satisfying suitable conditions (e.g., freely and properly), the quotient map p: X \to X/G is a covering map, hence open, ensuring the quotient topology respects local Euclidean structure and allows lifting of paths and homotopies through the fibers. This identification of fibers via openness preserves key invariants like homotopy groups above the base dimension, enabling applications in bundle theory and classification of manifolds.^[34]

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