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Preference numbers and funnel dimension. (English) Zbl 0725.54028

A new dimension function, funnel dimension (f-dim) is introduced, and the following properties are obtained (for its precise definition see the end of this review). (a) A one-point set X has f-dim X\(=0\). (c) If X is a finite \(T_ 1\)-space with more than one point, then f-dim X\(=1\). (d) If X is the unit interval, then f-dim X\(=1\). Theorem 1.5. If Y is a non- empty subset of X, then f-dim \(Y\leq f\)-dim X. Theorem 11.7. It holds that f-dim X\(=n\) if X is either n-dimensional simplex or n-dimensional Euclidean space. Theorem. For a separable metric space X we have ind \(X\leq f\)-dim \(X\leq 2 ind X+1\). Theorem 3.2. If X is separable metric, then f-dim \(X\leq d\)-dim X, where d-dim is introduced by E. Deak [Topics in Topol., Colloqu. Keszthely 1972, Colloquia Math. Soc. János Bolyai 8, 187-211 (1974; Zbl 0355.54007)]. Theorem 4.4. For a \(T_ 1\)- space X, f-dim \(X\leq 1\) iff X is embeddable in R. The authors say that mathematical economics suggests the following definition of f-dim. Definition 1.2. A funnel in a topological space X is a collection \({\mathcal F}=\{F_ t:\) \(t\in D\}\) of closed subsets, indexed by a dense subset D of real interval [0,1], satisfying (i) \(t<s\) implies \(F_ t\subset F_ s\), (ii) \(\cup \{F_ t:\) \(t\in D\}=X\), (iii) \(F_ t=\cap \{F_ s:\) \(s>t\), \(s\in D\}\) for each \(t\in D\). - Definition 1.3. The funnel dimension of a topological space X (f-dim X) is the least number \(n\geq 0\) for which there are \(n+1\) funnels \({\mathcal F}_ 1,...,{\mathcal F}_{n+1}\) which together constitute a subbase for the closed subsets of X. If no such number exists, we say that f-dim X\(=\infty\).

MSC:

54F45 Dimension theory in general topology
91B08 Individual preferences

Citations:

Zbl 0355.54007
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