Consistent Connectedness in Preferences

• Consistent connectedness characterizes how the topology of a choice set influences the behavioral properties of continuous binary relations.
• It demonstrates that on connected or 2-connected spaces, continuity enforces both completeness and transitivity in preference relations.
• The framework extends classic results by recasting preference axioms as outcomes of underlying topological conditions, informing empirical analysis in economics.

Searching arXiv for the specified paper and closely related context. Consistent connectedness denotes the two-way relationship between topological connectedness of a choice set and the behavioral properties of continuous binary relations defined on that set. In the formulation developed in "Topological Connectedness and Behavioral Assumptions on Preferences: A Two-Way Relationship" (Khan et al., 2018), connectedness is not merely a background regularity condition: under the standard continuity hypothesis for preferences, it becomes the critical topological condition governing whether consistency (transitivity), decisiveness (completeness), or both must hold. The paper presents this two-way relationship as the Eilenberg–Sonnenschein research program, giving synthetic characterizations of connectedness and $2$-connectedness in terms of universal behavioral implications for continuous relations, and deriving additional sufficient conditions for transitivity that do not require completeness (Khan et al., 2018).

1. Topological and order-theoretic framework

The central topological primitive is connectedness. A topological space $X$ is connected if it has no nontrivial separation: $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$ Equivalent formulations include: $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$ and

$$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$

In metric spaces, a standard equivalent is that if $f:X\to\mathbb{R}$ is continuous then $f(X)$ is an interval (Khan et al., 2018).

A component of $X$ is a maximal connected subset. The space is $k$-connected if it has at most $k$ components, with $X$0-connected equivalent to connectedness. The paper also uses path-connectedness and local connectedness in subsidiary arguments. A subset $X$1 is path-connected if for all $X$2 there exists a continuous $X$3 with $X$4 and $X$5. Local connectedness at a point means that every neighborhood contains a connected open neighborhood of that point (Khan et al., 2018).

The binary relation $X$6 is interpreted as a weak preference relation. Its strict and symmetric parts are

$X$7

The notation is $X$8 for $X$9, $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$0 for $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$1, and $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$2 for $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$3. The key behavioral properties are consistency or transitivity,

$$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$4

and decisiveness or completeness,

$$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$5

The paper also relies on acyclicity, quasi-transitivity, semi-transitivity, transitivity of $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$6, and negative transitivity of $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$7 (Khan et al., 2018).

Continuity is defined by the standard closed-sections and open-strict-sections condition: $$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$8 together with

$$X \text{ is connected} \;\Longleftrightarrow\; \neg\exists U,V\subset X \text{ open s.t. } U\cap V=\emptyset,\;U\cup V=X,\;U\neq\emptyset,\;V\neq\emptyset.$$9

For negatively transitive $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$0, open sections are equivalent to open graph, citing Border–Pattanaik–Riaz (1976) in the paper’s discussion (Khan et al., 2018).

2. The Eilenberg–Sonnenschein program

The paper organizes its contribution around two directions. The forward direction states that connectedness imposes behavioral constraints on continuous relations. On $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$1-connected spaces, continuity together with mild order properties forces completeness; for $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$2, it also forces transitivity. The backward direction reverses the implication: if every continuous relation with certain behavioral properties is complete, or complete and transitive, then the topology must be connected or $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$3-connected. Connectedness is therefore behaviorally indispensable under continuity (Khan et al., 2018).

This two-way structure is what the paper calls the Eilenberg–Sonnenschein research program. The significance of the formulation is that it treats connectedness not as an auxiliary topological assumption but as a property characterized by the universal validity of order-theoretic consequences. In that sense, connectedness and behavioral regularity become mutually defining under the continuity hypothesis.

A concise summary of the program appears in the paper’s synthesis. On connected and $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$4-connected spaces, any continuous relation that is semi-transitive with transitive indifference, or anti-symmetric, is necessarily complete, and for $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$5 also transitive. Conversely, if every continuous relation with those properties satisfies the corresponding behavioral conclusion, then the topology must be $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$6-connected or $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$7-connected. The paper records this as: $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$8 in the relevant cases, and also as the converse universal characterization: $$X \text{ connected} \;\Longleftrightarrow\; \text{the only clopen subsets are } \emptyset \text{ and } X,$$9

A plausible implication is that the ES program recasts classical preference axioms as topology-dependent consequences rather than independent primitives. That implication is consistent with the paper’s repeated claim that completeness and transitivity may be hidden in connectedness plus continuity.

3. Main equivalence theorems

The core of the theory is a set of six theorems, four of which are described as synthetic, complete characterizations of connectedness and its natural extensions (Khan et al., 2018).

Theorem 1 concerns $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$0-connectedness. Let $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$1 be a topological space with at least $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$2 components, and let $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$3 be a continuous, $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$4-non-trivial binary relation on $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$5. Then the following are equivalent: $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$6 is $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$7-connected; any transitive continuous relation is complete; any anti-symmetric continuous relation is complete; any continuous relation whose symmetric part $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$8 is transitive with connected sections is complete; and any semi-transitive continuous relation with transitive symmetric part is complete. The theorem identifies $$X \text{ connected} \;\Longleftrightarrow\; \text{every continuous } f:X\to\{0,1\} \text{ is constant}.$$9-connectedness through universal completeness consequences.

Theorem 2 strengthens these equivalences for $f:X\to\mathbb{R}$0. Under the same setting, the equivalents become: $f:X\to\mathbb{R}$1 is $f:X\to\mathbb{R}$2-connected; any transitive relation is complete; any anti-symmetric relation is complete and transitive; any relation whose symmetric part $f:X\to\mathbb{R}$3 is transitive with connected sections is complete and transitive; and any semi-transitive relation with transitive symmetric part is complete and transitive. The paper states that on connected or $f:X\to\mathbb{R}$4-connected spaces, continuity collapses indecision and inconsistency jointly: completeness and transitivity come “for free” from connectedness (Khan et al., 2018).

Theorem 3 characterizes $f:X\to\mathbb{R}$5-connectedness on quasi-ordered spaces, where quasi-ordered means that there exists at least one complete, anti-symmetric, continuous binary relation on $f:X\to\mathbb{R}$6. For any complete, continuous relation $f:X\to\mathbb{R}$7 on $f:X\to\mathbb{R}$8, the following are equivalent: $f:X\to\mathbb{R}$9 is $f(X)$0-connected; any anti-symmetric continuous relation is transitive; any continuous relation whose symmetric part $f(X)$1 is transitive with connected sections is transitive; and any semi-transitive continuous relation is transitive. This theorem ties the Eilenberg–Sonnenschein transitivity claims specifically to $f(X)$2-connectedness.

Theorem 4 provides a broader “portmanteau” characterization of connectedness on any quasi-ordered topological space $f(X)$3 with more than two elements. Its equivalent conditions include statements about strongly non-trivial and transitive relations with closed upper sections and negatively transitive strict parts, uniqueness or inversion of anti-symmetric non-trivial continuous relations, fragility of incomplete non-trivial transitive relations with closed sections, flimsiness of incomplete non-trivial transitive relations whose strict part has open sections, and the relation between continuous dual representation, strong separability, pseudo-transitivity, and closed covering sections. The theorem therefore connects connectedness to incompleteness phenomena and to representation-theoretic conditions (Khan et al., 2018).

4. Transitivity without completeness

Two further theorems isolate consequences that stem solely from connectedness, rather than from completeness. These results sharpen the logical interdependencies emphasized by Sen and place them in a topological setting (Khan et al., 2018).

Theorem 5 states that if $f(X)$4 is continuous and $f(X)$5 is connected, then semi-transitivity is equivalent to negative transitivity of $f(X)$6: $f(X)$7 It also yields

$f(X)$8

$f(X)$9

and

$X$0

If the sections of $X$1 are connected, then transitivity of $X$2 implies semi-transitivity,

$X$3

and transitivity of $X$4 becomes equivalent to transitivity of both $X$5 and $X$6: $X$7

Theorem 6 uses the Phragmén–Brouwer property. For separated open sets $X$8 with $X$9 and $k$0, the PBP requires a connected subset $k$1 separating $k$2 from $k$3. The property holds, for example, for convex subsets of Euclidean spaces and spheres of dimension $k$4. Under PBP and continuity of $k$5, path-connected upper sections of $k$6 together with $k$7 imply $k$8; path-connected lower sections together with $k$9 imply $k$0; and path-connected sections of $k$1 imply

$k$2

The paper emphasizes that these are parsimonious topological routes to transitivity that do not assume completeness (Khan et al., 2018).

This part of the theory matters because it shows that connectedness does more than force completeness under transitivity assumptions. It also reconstructs transitivity from weaker ingredients such as semi-transitivity and transitive indifference, provided continuity and suitable connectedness conditions on the domain or on relation sections are present.

5. Relation to classic results and illustrative constructions

The paper explicitly situates its results as generalizations of Eilenberg (1941), Sonnenschein (1965/67), Schmeidler (1971), and Sen (1969) (Khan et al., 2018). Eilenberg’s classic statement held that on a connected space, a complete, anti-symmetric, continuous relation is transitive, and he characterized orderability of connected spaces via disconnectedness of $k$3. Theorems 2 and 3 extend this by dropping completeness in Theorem 2(c), where completeness and transitivity become consequences, and by characterizing $k$4-connectedness as the topology required for universal transitivity claims on quasi-ordered spaces.

Sonnenschein’s classic results showed that under continuity and connectedness of $k$5, or connectedness of sections of $k$6, or path-connectedness plus PBP, completeness together with semi-transitivity implies full transitivity. Theorems 1 and 2 recover and strengthen this by making completeness a consequence of connectedness or by weakening connectedness to $k$7-connectedness while retaining transitivity implications. Theorem 6 recasts the PBP and path-connectedness conditions without completeness, deriving semi-transitivity from $k$8 and path connectivity.

Schmeidler’s result stated that on connected $k$9, continuity plus non-triviality and transitivity imply completeness. Theorem 1 recovers this as the implication from connectedness to universal completeness of transitive continuous relations and embeds it in a full equivalence characterizing $X$00-connectedness. Sen’s logical interdependencies among $X$01, $X$02, $X$03, $X$04, $X$05, and $X$06, classically stated under completeness, are reworked in Theorem 5 so that continuity plus connectedness carry much of the logical burden without completeness.

The paper also provides constructive examples that show the necessity of connectedness. On connected spaces, a continuous utility $X$07 induces a relation $X$08, with $X$09 given by $X$10 and $X$11 by $X$12. When level sets are intervals, $X$13 has connected sections, and by Theorem 2(d) the relation is complete and transitive. On convex subsets of $X$14, path-connected sections and transitivity of $X$15 allow Theorem 6(iii) to yield transitivity without completeness (Khan et al., 2018).

By contrast, the paper constructs disconnected-space examples where continuity coexists with failure of transitivity, completeness, or both. On $X$16, ordered cyclically across the three components, there is an anti-symmetric, complete, continuous relation that is not transitive. If $X$17 is disconnected with open partition $X$18, then

$X$19

is transitive and continuous but incomplete. On three components $X$20, the relation

$X$21

is continuous, incomplete, and non-transitive. These examples show that the forward guarantees of Theorems 1 and 2 fail exactly on disconnected spaces.

6. Empirical content, scope, and implications

A central interpretive point is the status of continuity. The paper describes continuity of preferences, understood as closed upper and lower contour sets together with open strict sections, as “non-testable” in finite datasets: no finite observation can falsify it. Yet continuity acquires substantive behavioral force when combined with connectedness (Khan et al., 2018). The paper illustrates this with cycling choices on $X$22 among $X$23, $X$24, and $X$25, choosing $X$26 over $X$27, $X$28 over $X$29, and $X$30 over $X$31. If one also insists on anti-symmetry and non-triviality, then Theorem 2(c) implies that on connected domains anti-symmetric continuous relations must be complete and transitive; the observed cycle therefore forces rejection of continuity.

This gives connectedness an empirical role in revealed preference analysis. On connected domains, continuity plus mild consistency properties force transitivity and or completeness. A plausible implication is that cycles observed in data on connected feasible sets can be read not only as failures of preference axioms but also as evidence against continuity, provided the auxiliary assumptions of the theorems are accepted.

The scope of the results is explicitly topological rather than metric or compactness-based. Theorems generally hold on arbitrary topological spaces; connectedness and $X$32-connectedness are independent of metrizability and compactness. Some backward directions require quasi-ordered spaces to avoid vacuity. The paper also notes that for finite spaces supporting anti-symmetric, complete, continuous relations, the quotient topology is discrete, and that non-Hausdorff spaces may fail to admit quasi-orders. In non-compact Hausdorff spaces, components coincide with quasi-components, although the paper works with components for the reported results (Khan et al., 2018).

The applications listed include microeconomic preference theory and utility representation, welfare and social choice, general equilibrium, and games. In Eilenberg–Debreu–Rader-type representation settings on connected, separable spaces, continuity plus anti-symmetry already force completeness and transitivity, so those latter properties function as hidden assumptions. On convex consumption sets in Walrasian existence proofs, or connected action spaces in games, the same hiddenness obtains. Conversely, models with indivisibilities and hence disconnected domains exhibit the failures that the theory predicts. The paper therefore presents consistent connectedness as a unifying topological principle linking continuous preferences, representation theory, and the behavioral structure of economic models (Khan et al., 2018).