MathDB

Let

\mathcal{T}_1

and

\mathcal{T}_2

be second-countable topologies on the set

E

. We would like to find a real function

\sigma

defined on

E \times 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 \in E

, the sets V_1(p,\varepsilon)\equal{}\{ x : \;\sigma(x,p)< \varepsilon \ \} \;(\varepsilon >0) form a neighborhood base of

p

with respect to

\mathcal{T}_1

, and the sets V_2(p,\varepsilon)\equal{}\{ x : \;\sigma(p,x)< \varepsilon \ \} \;(\varepsilon >0) form a neighborhood base of

p

with respect to

\mathcal{T}_2

. Prove that such a function

\sigma

exists if and only if, for any

p \in E

and

\mathcal{T}_i

-open set G \ni p \;(i\equal{}1,2) , there exist a

\mathcal{T}_i

-open set

G'

and a \mathcal{T}_{3\minus{}i}-closed set

F

with

p \in G' \subset F \subset G.

A. Csaszar

Problem Statement