Let T1 and T2 be second-countable topologies on the set E. We would like to find a real function σ defined on E×E such that 0 \leq \sigma(x,y) <\plus{}\infty, \;\sigma(x,x)\equal{}0 \ , \sigma(x,z) \leq
\sigma(x,y)\plus{}\sigma(y,z) \;(x,y,z \in E) \ , and, for any p∈E, the sets V_1(p,\varepsilon)\equal{}\{ x : \;\sigma(x,p)< \varepsilon \ \} \;(\varepsilon >0) form a neighborhood base of p with respect to T1, and the sets V_2(p,\varepsilon)\equal{}\{ x : \;\sigma(p,x)< \varepsilon \ \} \;(\varepsilon >0) form a neighborhood base of p with respect to T2. Prove that such a function σ exists if and only if, for any p∈E and Ti-open set G \ni p \;(i\equal{}1,2) , there exist a Ti-open set G′ and a \mathcal{T}_{3\minus{}i}-closed set F with p∈G′⊂F⊂G.
A. Csaszar functiontopologyadvanced fieldsadvanced fields unsolved