Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let T be the family of open subsets of a topological space (not necessarily Hausdorfi or even T0).  prove that if T has a base of cardinality • „, ‚ • „ < 2 ‚ , ‚ strong limit of coflnality @0, then T has cardinality • „ or ‚ 2 ‚ . This is our main conclusion (21). In Theorem 2  prove it under some set theoretic assumption, which is clear when ‚ = „; then  eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next  prove that the simplest examples are the basic ones; they occur in every example (for ‚ =@0 this fulflll a promise from [Sh 454]). The main result for the case ‚ =@0 was proved in [Sh 454].
['Saharon Shelah']
I am writing a research paper on a idea Cardinalities of topologies with small base
[{"from": "human", "value": "I am writing a research paper on a idea Cardinalities of topologies with small base"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let T be the family of open subsets of a topological space (not necessarily Hausdorfi or even T0).  prove that if T has a base of cardinality \u2022 \u201e, \u201a \u2022 \u201e < 2 \u201a , \u201a strong limit of coflnality @0, then T has cardinality \u2022 \u201e or \u201a 2 \u201a . This is our main conclusion (21). In Theorem 2  prove it under some set theoretic assumption, which is clear when \u201a = \u201e; then  eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next  prove that the simplest examples are the basic ones; they occur in every example (for \u201a =@0 this fulflll a promise from [Sh 454]). The main result for the case \u201a =@0 was proved in [Sh 454]."}]