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Subspace topology

In , the subspace topology (also called the relative or induced topology) is the canonical way to endow a of a with its own topology, derived directly from the topology of the ambient space. For a (X, \mathcal{T}) and any Y \subseteq X, the subspace topology \mathcal{T}_Y on Y is defined as the collection of all sets of the form U \cap Y, where U \in \mathcal{T}. This construction ensures that the subspace (Y, \mathcal{T}_Y) inherits the intuitive notion of and closedness from X, making it a in its own right. The subspace topology is foundational to , as it facilitates the study of subsets without altering their embedding in the larger space, and it underpins many advanced concepts such as quotient spaces and product topologies. Key properties include the continuity of the i: Y \to X, which is always continuous with respect to \mathcal{T}_Y and \mathcal{T}, and the fact that restrictions of continuous functions from X to Y remain continuous. If \mathcal{B} is a basis for \mathcal{T}, then \{B \cap Y \mid B \in \mathcal{B}\} serves as a basis for \mathcal{T}_Y. Moreover, certain separation axioms like are hereditary, meaning a subspace of a is also Hausdorff. In metric spaces, this topology aligns with the standard metric-induced topology on subsets, reinforcing its role in both abstract and concrete settings. Subspaces are instrumental in theorems such as the pasting lemma, which allows gluing continuous functions defined on disjoint open or closed subspaces that agree on their intersection to form a global continuous function. They also preserve local compactness when the subset is open or closed in a locally compact Hausdorff space, and they enable unique extensions of continuous maps under certain conditions, such as from a dense subspace to its closure in Hausdorff targets. These features highlight the subspace topology's utility in decomposing complex spaces into manageable components for analysis.

Definition and Construction

Formal Definition

Let (X, \tau) be a and A \subseteq X a . The subspace topology (also called the relative topology or induced topology) on A, denoted \tau_A, is the collection of all sets of the form U \cap A where U \in \tau. The topological space (A, \tau_A) is called a subspace of (X, \tau). To verify that \tau_A is indeed a on A, note that it satisfies the three topology axioms inherited from \tau. First, the and A are in \tau_A, since \emptyset = \emptyset \cap A with \emptyset \in \tau, and A = X \cap A with X \in \tau. Second, \tau_A is closed under arbitrary unions: if \{V_\alpha \cap A \mid \alpha \in I\} is a collection of sets in \tau_A, then \bigcup_{\alpha \in I} (V_\alpha \cap A) = \left( \bigcup_{\alpha \in I} V_\alpha \right) \cap A with \bigcup_{\alpha \in I} V_\alpha \in \tau. Third, \tau_A is closed under finite s: for V_1 \cap A, \dots, V_n \cap A \in \tau_A, their intersection is (V_1 \cap \cdots \cap V_n) \cap A with V_1 \cap \cdots \cap V_n \in \tau. The subspace topology \tau_A arises from the inclusion map i: A \to X defined by i(a) = a for all a \in A, which is continuous with respect to \tau_A and \tau. More precisely, \tau_A is the initial topology (or coarsest topology) on A that makes i continuous, satisfying the following universal property: for any topological space W and continuous map f: W \to X with f(W) \subseteq A, there exists a unique continuous map f': W \to (A, \tau_A) such that i \circ f' = f. This characterizes \tau_A uniquely among all topologies on A for which i is continuous.

Basis and Subbasis for Subspaces

In a topological space (X, \tau), if \mathcal{B} is a basis for \tau, then the collection \mathcal{B}_A = \{ B \cap A \mid B \in \mathcal{B} \} forms a basis for the subspace topology \tau_A on a subset A \subseteq X. To verify this, first note that \mathcal{B}_A covers A: for any a \in A, there exists B \in \mathcal{B} such that a \in B, so a \in B \cap A \in \mathcal{B}_A. Next, \mathcal{B}_A satisfies the basis intersection property relative to \tau_A: consider U, V \in \mathcal{B}_A with a \in U \cap V. Then U = B_1 \cap A and V = B_2 \cap A for some B_1, B_2 \in \mathcal{B}, so a \in (B_1 \cap B_2) \cap A. Since \mathcal{B} is a basis for \tau, there exists B_3 \in \mathcal{B} such that a \in B_3 \subseteq B_1 \cap B_2, and thus B_3 \cap A \in \mathcal{B}_A with a \in B_3 \cap A \subseteq (B_1 \cap B_2) \cap A = U \cap V. Moreover, every open set in \tau_A is a union of elements from \mathcal{B}_A, as \tau_A = \{ U \cap A \mid U \in \tau \} and each U \in \tau is a union of basis elements from \mathcal{B}. A key characterization follows: a set V \subseteq A is open in \tau_A if and only if for every a \in V, there exists B \in \mathcal{B} such that a \in B \cap A \subseteq V. This provides a constructive for openness in the using the ambient basis. Similarly, if \mathcal{S} is a subbasis for \tau, then \mathcal{S}_A = \{ S \cap A \mid S \in \mathcal{S} \} is a subbasis for \tau_A, in the sense that the topology generated by \mathcal{S}_A—consisting of all unions of finite intersections of elements from \mathcal{S}_A—coincides with \tau_A. The finite intersections of elements from \mathcal{S}_A form a basis for \tau_A, mirroring the role of \mathcal{S} in generating \tau. For example, in the \mathbb{R}^n with the standard topology, where open balls form a basis, the subspace topology on A \subseteq \mathbb{R}^n has as a basis the sets B(a, r) \cap A for a \in A and r > 0, where B(a, r) is the open ball centered at a with radius r. This restricted basis captures the local structure of A inherited from \mathbb{R}^n.

Terminology and Examples

Key Terminology

In the context of subspace topology, the ambient space refers to the original X from which a subset A \subseteq X is drawn to form a . This term emphasizes the of the within the larger structure of X, where properties of A are analyzed relative to the of X. The subspace topology on A, denoted \tau_A, is also known as the relative topology or induced topology, highlighting its derivation directly from the open sets of the ambient space X. Specifically, \tau_A = \{ U \cap A \mid U \in \tau \}, where \tau is the on X, ensuring that the topology on A inherits the structure of X without introducing extraneous openness. A subset V \subseteq A is defined as open in A if it belongs to \tau_A, meaning V = U \cap A for some U in the ambient space X, and similarly for closed in A using closed sets in X. This relative notion distinguishes openness or closedness with respect to \tau_A from absolute openness or closedness in X. The term subspace properly denotes the ordered pair (A, \tau_A), which equips the set A with its induced topology, rather than the set A alone, underscoring that the topological structure is integral to the object. This pairing ensures that (A, \tau_A) functions as a in its own right. The concept of subspace topology builds on foundational ideas in point-set introduced by in his 1914 work Grundzüge der Mengenlehre, and the specific terminology was formalized in mid-20th-century texts such as those by Kuratowski and others.

Illustrative Examples

A fundamental example of a subspace topology arises in the \mathbb{R} equipped with its standard topology, generated by open intervals. Consider the closed interval Y = [0,1] as a subspace of \mathbb{R}. The open sets in this subspace are of the form U \cap [0,1], where U is open in \mathbb{R}. For instance, near the endpoint 0, a typical basis element is [0, \epsilon) for \epsilon > 0, which is the intersection of the open interval (-\delta, \epsilon) with [0,1] for some \delta > 0. This topology on [0,1] is not the same as the standard topology on \mathbb{R} restricted to open intervals within [0,1], as sets like [0, \epsilon) are open in the subspace but not in \mathbb{R}. Another illustrative case is a discrete subspace. Let X be any topological space, and let A \subseteq X be such that every singleton \{a\} for a \in A is open in X. Then, in the subspace topology on A, each singleton \{a\} = U \cap A where U = \{a\} is open in X, so the subspace topology on A is discrete, meaning every subset of A is open. A concrete realization occurs with the integers \mathbb{Z} as a subspace of \mathbb{R} under the standard topology: each \{n\} = (n - 0.5, n + 0.5) \cap \mathbb{Z} is open in the subspace, yielding the discrete topology on \mathbb{Z}. This contrasts with non-discrete subspaces like the rationals \mathbb{Q} in \mathbb{R}, where no non-empty proper subsets without limit points are open. The subspace topology on the irrationals \mathbb{R} \setminus \mathbb{Q}, denoted \mathbb{P}, inherited from the standard topology on \mathbb{R}, provides insight into more pathological structures. Open sets in \mathbb{P} are intersections of open intervals in \mathbb{R} with \mathbb{P}, such as (a, b) \cap \mathbb{P} for rationals a < b. These sets are both open and closed (clopen) in \mathbb{P} because their complements in \mathbb{P} are unions of similar clopen sets. Thus, \mathbb{P} admits a basis of clopen sets, making it zero-dimensional. However, it is not discrete, as singletons are not open; any open set in \mathbb{P} is infinite and dense in itself due to the density of irrationals in \mathbb{R}. This topology highlights how subspace inheritance can produce spaces that are totally disconnected yet non-discrete. The offers a minimal example involving finite topologies. Consider X = \{0,1\} with the topology \tau = \{\emptyset, \{1\}, \{0,1\}\}, known as the Sierpiński topology where \{0\} is closed but not open. Now take the subspace Y = \{0\}. The subspace topology on Y consists of sets U \cap \{0\} for U \in \tau. The only possibilities are \emptyset \cap \{0\} = \emptyset and \{1\} \cap \{0\} = \emptyset, along with \{0,1\} \cap \{0\} = \{0\}, yielding \tau_Y = \{\emptyset, \{0\}\}—the indiscrete (trivial) topology on Y. This illustrates how the subspace topology can coarsen dramatically, making even a space non-discrete. Finally, the subspace topology \tau_Y on a Y \subseteq X is the coarsest topology making the i: (Y, \tau_Y) \to (X, \tau_X) . To see this, note that of i requires i^{-1}(V) = V \cap Y to be open in Y for every open V \in \tau_X. Thus, \tau_Y is the topology generated by taking these V \cap Y as a subbasis. Any coarser topology on Y would fail to include some V \cap Y as open, violating . Conversely, any finer topology \sigma \supseteq \tau_Y would still have all V \cap Y open, preserving of i. This establishes \tau_Y as the minimal such topology.

Fundamental Properties

Algebraic Properties

In the subspace topology on a subset A of a X, the interior and operators relative to A are determined by their counterparts in X. Specifically, for any V \subseteq A, the interior of V in the subspace topology on A, denoted \operatorname{int}_A(V), is given by \operatorname{int}_A(V) = \operatorname{int}_X(V) \cap A, where \operatorname{int}_X(V) is the interior of V in X. Similarly, the of V in A, denoted \operatorname{cl}_A(V), satisfies \operatorname{cl}_A(V) = \operatorname{cl}_X(V) \cap A, with \operatorname{cl}_X(V) the in X. These relations follow directly from the definition of the subspace topology, where open sets in A are intersections of open sets in X with A, and closed sets in A are intersections of closed sets in X with A. The subspace topology inherits the algebraic structure of unions and intersections from X. Arbitrary unions of sets open in the subspace topology on A remain open in A: if \{V_i\}_{i \in I} are open in A, then each V_i = U_i \cap A for some open U_i in X, and \bigcup_{i \in I} V_i = \left( \bigcup_{i \in I} U_i \right) \cap A, where \bigcup_{i \in I} U_i is open in X. Likewise, finite intersections of subspace-open sets are subspace-open: for V_1 = U_1 \cap A and V_2 = U_2 \cap A with U_1, U_2 open in X, V_1 \cap V_2 = (U_1 \cap U_2) \cap A, and U_1 \cap U_2 is open in X. This confirms that the collection of subspace-open sets forms a topology on A. A set V \subseteq A is open in the subspace topology if and only if V = U \cap A for some open set U in X. Regarding complements within the subspace, for B \subseteq A, the relative complement A \setminus B is open in the topology on A B is closed in A. This equivalence holds because A \setminus B open in A means there exists an open U in X such that A \setminus B = U \cap A, implying B = A \cap (X \setminus U) where X \setminus U is closed in X, so B is closed in A. Although A may be closed in X, subsets of A that are closed in the subspace topology need not be closed in X. For instance, if A is closed in X, a set closed in A takes the form C \cap A for some closed C in X, but C \cap A may fail to contain all limit points of itself in X that lie outside A. This distinction highlights how the subspace topology can introduce new closed sets not present in the ambient space.

Continuity in Subspaces

In the context of subspace topology, continuity of a f: A \to Y, where A is a of a (X, \tau) and (Y, \sigma) is a , is defined relative to the topology \tau_A on A. Specifically, f is if and only if for every W \in \sigma, the preimage f^{-1}(W) is open in A with respect to \tau_A. This relative notion ensures that respects the induced structure on the subspace without requiring openness in the ambient space X. When the is also a , say B is a of a (Z, \rho), a f: A \to B is continuous if and only if for every V in B (with subspace topology \rho_B), the preimage f^{-1}(V) is open in A. Since open sets in B are of the form U \cap B for U \in \rho, this condition is equivalent to f^{-1}(U \cap B) being open in A whenever U is open in Z. This characterization highlights how subspace topologies preserve the preimage criterion for across induced structures. The i: A \to X, defined by i(a) = a for all a \in A, is always continuous by the construction of the subspace topology \tau_A = \{U \cap A \mid U \in \tau\}. This follows directly from the fact that for any open U \in \tau, i^{-1}(U) = U \cap A, which is open in A. Thus, the subspace topology is the coarsest topology on A making the continuous. For local continuity, consider a . The map g is at a (with A a of X) if and only if the restriction g|_A: A \to Y is continuous at p relative to the subspace topology on A. In other words, for every open neighborhood V of g(p) in Y, there exists an open neighborhood W of p in X such that g(W) \subseteq V and W \cap A is a neighborhood of p in A. This equivalence ties pointwise continuity in the ambient space to relative continuity in the . A precise formulation of continuity for f: (A, \tau_A) \to (Y, \sigma) is given by the following equivalence: f \text{ is continuous} \iff \forall V \in \sigma, \ f^{-1}(V) = U \cap A \text{ for some } U \in \tau. This condition underscores the interplay between the subspace and ambient topologies in determining continuity.

Preservation of Topological Properties

Inherited Properties

Subspaces inherit several key topological properties from the ambient space, ensuring that the subspace topology behaves consistently with respect to these invariants. One fundamental property is compactness: if a subset A of a topological space X is compact in X, then A equipped with the subspace topology \tau_A is also compact. To see this, consider an open cover \{U_i \cap A \mid i \in I\} of A in \tau_A, where each U_i is open in X. This collection lifts to the cover \{U_i \mid i \in I\} of A in X, which admits a finite subcover \{U_{i_1}, \dots, U_{i_n}\}. The corresponding \{U_{i_k} \cap A \mid k = 1, \dots, n\} then forms a finite subcover in \tau_A. Connectedness is similarly preserved in the subspace topology. A subset A \subseteq X is connected in the subspace topology it is connected as a subset of X, meaning A cannot be expressed as the union of two disjoint nonempty relatively open sets in \tau_A. This equivalence holds because any disconnection in \tau_A would correspond to a disconnection of A in X via the open sets generating the relatively open sets, ensuring that connected subsets remain connected under the induced topology. Hausdorff separation is another inherited property: every subspace of a Hausdorff space is Hausdorff. For distinct points y_1, y_2 \in Y \subseteq X, where X is , there exist disjoint open sets U_1, U_2 \subseteq X with y_1 \in U_1 and y_2 \in U_2. The intersections U_1 \cap Y and U_2 \cap Y are then disjoint open sets in the subspace topology separating y_1 and y_2. Metrizability also passes to subspaces. If X is a metrizable space with d, then any Y \subseteq X is metrizable via the restricted d|_Y(a, b) = d(a, b) for a, b \in Y, which induces the subspace topology on Y. This follows directly from the fact that the open balls in the restricted coincide with the intersections of open balls in X with Y. Finally, first-countability and second-countability are hereditary properties. For first-countability, if X has a countable local basis at each point x \in X, then for any y \in Y \subseteq X, the countable collection \{B_n \cap Y \mid B_n \text{ is a local basis at } y \text{ in } X\} serves as a countable local basis at y in the subspace topology. Similarly, second-countability inherits via the restriction of a countable basis for X to Y, yielding a countable basis for \tau_Y consisting of intersections with the original basis elements.

Limitations and Counterexamples

While the real line \mathbb{R} with the standard is locally compact, its consisting of the rational numbers \mathbb{Q} is not locally compact. This follows from the fact that every compact subset of \mathbb{Q} is nowhere dense in \mathbb{R}, so no neighborhood of any point in \mathbb{Q} contains a compact set with non-empty interior in the . A classic example of a that is connected but not path-connected is the in \mathbb{R}^2, defined as the set S = \{(x, \sin(1/x)) \mid 0 < x \leq 1\} \cup \{(0, y) \mid -1 \leq y \leq 1\} equipped with the . The set S is connected because it cannot be written as a disjoint union of two non-empty open sets in the , but there is no continuous path from a point on the vertical segment at x=0 to a point on the sine curve portion, as any such path would require traversing infinitely many oscillations in finite time. In non-Hausdorff spaces, subspaces often inherit the lack of separation properties; for instance, the line with double point—a quotient of the real line where one point is duplicated—is T_1 but not Hausdorff, and any subspace containing both origins fails to separate those points with disjoint open neighborhoods. This illustrates how weaker separation axioms like T_1 do not prevent subspaces from failing stronger ones like Hausdorffness when the ambient space lacks them. Second-countable spaces exhibit countability failures in their subspaces: no second-countable space, such as \mathbb{R}, can contain an uncountable discrete subspace, since a discrete uncountable subset would demand uncountably many pairwise disjoint non-empty open sets, exceeding the countable basis. The Knaster–Kuratowski fan provides another pathological example: this connected subspace of the plane, constructed from the by drawing line segments to an point and including rational endpoints in a dispersed manner, becomes totally disconnected upon removal of the . Specifically, the fan is the union of segments from points in the C \subset [0,1] \times \{0\} to the (1/2, 1), with the subspace topology; removing the yields countably many connected components corresponding to rational dispersion points and uncountably many singletons. These counterexamples underscore the coarseness of the subspace topology, which induces the minimal collection of open sets making the continuous, often failing to preserve finer local or global structures and motivating alternative constructions like quotient topologies to achieve desired properties.