## 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

funnel dimension

Zbl 0355.54007
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