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Separated sets

In topology, separated sets are two nonempty subsets A and B of a topological space X such that A \cap B = \emptyset, A \cap \overline{B} = \emptyset, and \overline{A} \cap B = \emptyset, where \overline{A} and \overline{B} denote the closures of A and B, respectively; this means the sets are disjoint and neither contains limit points of the other.^[1]^[2] This concept is central to the study of connectedness in topological spaces. A space X is defined as disconnected if it can be expressed as the union of two nonempty separated sets whose union is X, and conversely, connected if no such separation exists.^[1]^[2] For subspaces, a subset Y \subseteq X is connected if it admits no separation into two nonempty relatively separated sets within the subspace topology.^[3] Key properties include the fact that the image of a connected set under a continuous function is connected, preserving the absence of separations.^[1] Beyond connectedness, separated sets play a role in higher separation axioms. For instance, in a completely normal (or T5) space, any two separated sets can be separated by disjoint open neighborhoods, extending the T4 (normal) axiom from disjoint closed sets.^[4] This framework applies to metric spaces, where the definition aligns with the topological one, and has implications for properties like path-connectedness and local connectedness in more structured spaces.^[1]

Fundamental Concepts

Definition

In a topological space X, two subsets A and B are said to be separated if \overline{A} \cap B = \emptyset and \overline{B} \cap A = \emptyset, where \overline{A} and \overline{B} denote the closures of A and B, respectively.^[4]^[5] This condition implies that A \cap B = \emptyset, as any intersection point would belong to both closures.^[6] The closures play a central role in this definition by incorporating limit points: a point lies in the closure of a set if it is either in the set or is a limit point of sequences (or nets) from that set. Thus, separated sets ensure that no point of one set belongs to the other or serves as a limit point of the other, preventing any topological "adherence" between them.^[7] Intuitively, this means the sets are topologically isolated from each other, with no overlap even in their accumulated boundaries.^[2] For example, in the real line \mathbb{R} equipped with the standard topology, the sets (-\infty, 0] and [1, \infty) are separated. Here, the closure of (-\infty, 0] is itself, and the closure of [1, \infty) is itself; neither closure intersects the other set, as the gap between 0 and 1 ensures complete separation.^[8]

Equivalent Formulations

A standard equivalent formulation of separated sets relies on neighborhoods. Two subsets A and B of a topological space X are separated if there exist open sets U, V \subseteq X such that A \subseteq U, B \subseteq V, U \cap B = \emptyset, and V \cap A = \emptyset. Another characterization uses filters. Let A and B be subsets of a topological space X. An ultrafilter \mathcal{F} on A converges to a point x \in X if every open neighborhood of x contains a member of \mathcal{F}. Then A and B are separated if no ultrafilter on A converges to any point in B, and no ultrafilter on B converges to any point in A. This holds because a point x lies in the closure of A if and only if there exists an ultrafilter on A that converges to x.^[9] The closure-based definition from the previous section is equivalent to the neighborhood formulation. To see that the closure condition implies the neighborhood condition, note that if \overline{A} \cap B = \emptyset and A \cap \overline{B} = \emptyset, then U = X \setminus \overline{B} is open, contains A (since A \cap \overline{B} = \emptyset), and satisfies U \cap B = \emptyset. Similarly, V = X \setminus \overline{A} is open, contains B, and satisfies V \cap A = \emptyset. Conversely, suppose there exist open U \supseteq A with U \cap B = \emptyset and open V \supseteq B with V \cap A = \emptyset. Assume for contradiction that p \in \overline{A} \cap B. Then V, an open neighborhood of p, intersects A, so V \cap A \neq \emptyset, contradicting the assumption. The argument for A \cap \overline{B} = \emptyset is symmetric. The filter-based formulation follows directly from the ultrafilter characterization of closure points.

Properties and Characterizations

Closure-Based Properties

In a topological space X, if subsets A and B are separated—meaning \overline{A} \cap \overline{B} = \emptyset—then both A and B are open (and closed) in the subspace topology on S = A \cup B. To see this, note that the closure of A in S is \overline{A} \cap S = \overline{A} \cap (A \cup B) = (\overline{A} \cap A) \cup (\overline{A} \cap B) = A \cup \emptyset = A, so A is closed in S; the argument for B is symmetric. Since A and B are disjoint and S = A \cup B, the complement of A in S is B, which is closed, making A open in S, and likewise for B.^[10] Separatedness also exhibits stability under finite unions relative to a fixed set. Suppose B \subseteq X is separated from each of finitely many pairwise disjoint sets A_1, \dots, A_n \subseteq X. Let A = \bigcup_{i=1}^n A_i. Then \overline{A} = \bigcup_{i=1}^n \overline{A_i}, so \overline{A} \cap B = \bigcup_{i=1}^n (\overline{A_i} \cap B) = \bigcup_{i=1}^n \emptyset = \emptyset; similarly, B \cap \overline{A} = \emptyset. Thus, A remains separated from B. This property underscores the robustness of separatedness with respect to finite amalgamations.^[10] When one set is compact and the other closed, separatedness yields stronger structural implications. In metric spaces, if A is compact and B is closed with \overline{A} \cap \overline{B} = \emptyset, the distance \inf \{ d(a,b) \mid a \in A, b \in B \} is positive. To see this, suppose the infimum is zero. Then there exist sequences a_n \in A and b_n \in B with d(a_n, b_n) \to 0. Since A is compact, there is a subsequence a_{n_k} \to a \in A. Along this subsequence, b_{n_k} \to a, and since B is closed, a \in B. But then a \in \overline{A} \cap B = \emptyset, a contradiction. More generally, in topological terms, such pairs can be separated by disjoint open neighborhoods in regular spaces: for each x \in A, there exist disjoint opens U_x \ni x and V_x \ni B; a finite subcover of the U_x yields open U \ni A and V = \bigcap V_x \ni B with U \cap V = \emptyset. In uniform spaces, an analogue ensures the sets are "uniformly separated" via entourages excluding close pairs.^[11]^[12]

Distinguishability of Points

In a topological space, two distinct points x and y are said to be topologically distinguishable if the singleton sets \{x\} and \{y\} form a pair of separated sets, meaning \{x\} \cap \overline{\{y\}} = \emptyset and \{y\} \cap \overline{\{x\}} = \emptyset.^[5] This condition ensures that neither point lies in the closure of the singleton consisting of the other point. This distinguishability is characterized by the existence of open neighborhoods U of x not containing y and V of y not containing x.^[13] Equivalently, the neighborhood systems of x and y differ, as there is at least one open set that contains exactly one of the two points.^[14] A topological space is T0, also known as Kolmogorov, if every pair of distinct points is topologically distinguishable in this sense.^[14] In such spaces, the topology provides a minimal level of separation between points based on their open neighborhoods. For example, in the indiscrete topology on a set with more than one point, where the only open sets are the empty set and the entire space, no two distinct points are distinguishable, as every nonempty open set contains both points and their closures are the whole space.^[13] Conversely, in the discrete topology, every subset is open, so singletons are closed and every pair of distinct points is distinguishable, with \{x\} itself serving as an open neighborhood of x excluding y.^[2]

Relations to Separation Axioms

T1 spaces

In topology, a T₁ space is defined as a topological space X in which, for any two distinct points x, y \in X, there exists an open neighborhood U of x that does not contain y and an open neighborhood V of y that does not contain x.^[15] This property ensures that points can be distinguished by their open neighborhoods in a symmetric manner, building on the notion of distinguishability of points where one point can be isolated from another via open sets.^[16] A key characterization of T₁ spaces is that every singleton set \{x\} is closed. To see this equivalence, suppose X is T₁. For a fixed x \in X, consider the complement X \setminus \{x\}. For each y \in X \setminus \{x\}, there exists an open neighborhood V_y of y not containing x. The union \bigcup_{y \neq x} V_y = X \setminus \{x\} is then open, so \{x\} is closed. Conversely, if every singleton is closed, then for distinct x, y \in X, the complement X \setminus \{y\} is open and contains x but not y, and similarly X \setminus \{x\} is open containing y but not x. Thus, the T₁ axiom holds.^[15]^[16] In T₁ spaces, singletons are separated sets, since \{x\} \cap \overline{\{y\}} = \emptyset and \overline{\{x\}} \cap \{y\} = \emptyset. The term "separated space" in topology typically refers to Hausdorff (T₂) spaces, where distinct points have disjoint open neighborhoods, a stronger condition. This terminology arose in early 20th-century developments of separation axioms, with variations in usage; for example, in French texts, "espace séparé" often means T₂, while in algebraic geometry, "separated" refers to the diagonal being closed, analogous to the T₁ condition.^[16]^[17]

Connection to Hausdorff Spaces

A topological space X is said to be Hausdorff, or to satisfy the T_2 separation axiom (also known as a separated space), if for every pair of distinct points x, y \in X, there exist disjoint open neighborhoods U of x and V of y, that is, U \cap V = \emptyset.^[18] This condition ensures that the singletons \{x\} and \{y\} can be openly separated, which in turn implies that they are separated sets in the sense that neither lies in the closure of the other. The Hausdorff axiom builds upon and strengthens the T_1 axiom. A space is T_1 if every singleton set is closed, meaning that for any distinct points x, y \in X, x \notin \overline{\{y\}} and y \notin \overline{\{x\}}, so \{x\} \cap \overline{\{y\}} = \emptyset and \overline{\{x\}} \cap \{y\} = \emptyset, making the singletons separated sets.^[15] However, while T_1 guarantees this closure-based separation for points, the Hausdorff property imposes the stricter requirement of disjoint open neighborhoods around them, enabling a more robust distinguishability. A classic example illustrating a T_1 space that fails to be Hausdorff is the line with double origin. This space is formed by taking the real line \mathbb{R} and adjoining a second origin o' to the origin o, with the topology generated by the usual open sets of \mathbb{R} together with sets of the form (a, b) \setminus \{o\} \cup \{o'\} for intervals (a, b) containing $0. Singletons remain closed, so distinct points—including o and o'—are separated sets, but no disjoint open neighborhoods exist to separate o and o'.^[19] In Hausdorff spaces, the separation concept extends beyond points: any two disjoint closed sets A and B are separated sets, since \overline{A} = A and \overline{B} = B, ensuring A \cap \overline{B} = A \cap B = \emptyset and \overline{A} \cap B = \emptyset.^[20] This holds generally in any topological space but underscores how the Hausdorff condition preserves the closure-based separation for closed subsets while enhancing pointwise distinguishability through open sets.

Broader Topological Connections

Relation to Connected Spaces

In topology, separated sets play a fundamental role in characterizing disconnected spaces. Specifically, a topological space X is disconnected if it can be expressed as the union of two nonempty separated sets whose union is X.^[21] This condition implies that X admits a separation into disjoint clopen subsets whose closures do not intersect each other, preventing the space from being connected. Conversely, a space X is connected if and only if it cannot be written as the union of two nonempty separated sets.^[21] The connected components of a topological space further illustrate the interplay between separated sets and connectedness. The connected components of X are the maximal connected subspaces, and any two distinct connected components are pairwise separated, meaning their closures in X are disjoint.^[21] This separation ensures that the components form a partition of X into disjoint closed sets, each of which is connected but cannot be extended without merging with another component. In particular, the closure of a connected component is itself connected and closed.^[21] Quasicomponents provide another decomposition related to separated sets, defined as the intersection of all clopen sets containing a given point x \in X. Unlike connected components, quasicomponents may not be connected, but any two distinct quasicomponents of X are separated in X, with their closures disjoint.^[22] This property holds in any topological space and underscores the global separation enforced by clopen sets, which are both open and closed. In compact Hausdorff spaces, quasicomponents coincide with connected components, both being single points in totally disconnected examples.^[21] A classic example highlighting these relations is the space of rational numbers \mathbb{Q} with the subspace topology inherited from \mathbb{R}. Here, the connected components are precisely the singletons \{q\} for each q \in \mathbb{Q}, as any subset of \mathbb{Q} with more than one point is disconnected and can be separated by open sets in \mathbb{R}.^[23] These singleton components are pairwise separated, since the closure of \{q\} in \mathbb{Q} is itself and disjoint from other points, reflecting the total disconnectedness of \mathbb{Q}. The quasicomponents also coincide with these singletons, reinforcing the separation across the space.^[22]

Applications in Quotient Spaces

In quotient spaces, separated sets play a key role in determining how topological properties transfer under the identification induced by an equivalence relation \sim on a topological space X. The quotient space X/{\sim} is equipped with the quotient topology, where the canonical projection p: X \to X/{\sim} is continuous and surjective, and a subset U \subseteq X/{\sim} is open if and only if p^{-1}(U) is open in X. A subset A \subseteq X is saturated with respect to p if A = p^{-1}(p(A)), meaning A is a union of entire equivalence classes $$. If two saturated sets A and B in X are separated—i.e., \overline{A} \cap B = \emptyset and \overline{B} \cap A = \emptyset—then their images p(A) and p(B) are separated in X/{\sim}. Conversely, separated sets in the quotient always lift to separated sets in X. If C, D \subseteq X/{\sim} are separated, let A = p^{-1}(C) and B = p^{-1}(D); then A and B are disjoint since C and D are. Moreover, if x \in \overline{A} \cap B, then p(x) \in \overline{p(A)} \cap p(B) \subseteq \overline{C} \cap D = \emptyset, a contradiction, and similarly for \overline{B} \cap A. Thus, the equivalence relation respects separation in the sense that the projection p preserves and reflects the separated property bidirectionally for appropriate sets. A related application concerns when the quotient X/{\sim} is Hausdorff, which requires that distinct points in X/{\sim} (i.e., inequivalent classes) can be separated by disjoint open sets. This holds if and only if the graph of \sim—the set \{(x,y) \in X \times X \mid x \sim y\}—is closed in X \times X, assuming X is Hausdorff and p is an identification map (continuous, surjective, and U \subseteq X/{\sim} open iff p^{-1}(U) open in X). The closed graph ensures that separated points in X remain distinguishable post-identification, preventing "accidental" gluings that merge closures.^[24]

References

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Below is a merged summary of separated sets and related topics in Munkres' *Topology* (2nd Edition), consolidating all information from the provided segments into a comprehensive response. To maximize detail and clarity, I will use a table in CSV format for key concepts, theorems, and page references, followed by a narrative summary that ties everything together. The response retains all mentioned details, including page numbers, theorems, and URLs, while addressing the lack of an explicit definition for "separated sets" and focusing on related topological properties.
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A T_1-space is a topological space fulfilling the T1-separation axiom: For any two points x,y in X there exists two open sets U and V such that x in U and y ...
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### Definition of T1 Space
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For example, the line with double origin is. Page 3. 13. Metrization of ... not a Hausdorff space since the point ˜0 cannot be separated by open sets ...
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Mar 22, 2013 · separated. Definition Suppose A A and B B are subsets of a topological space ... In [2] , separated sets are called strongly disjoint sets.
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Thus, no set containing more than one point can be connected. Hence, the only components are sets containing a single point. 4. If X has a finite number of ...
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In the plane R2 identify the x-axis to a point and give the resulting set X the quotient topology. (a) Is X Hausdorff? (b) Is X regular? (c) Does the ...