Suppose \(X\) admits a family \(\{U_r\}_{r \in \mathbb{Q}}\) of nested neighbourhoods. Define\(f: X \rightarrow [0,1]\) by \(f(x) =inf \mathbb{Q}(x)\). Verify the following:

(1) \(f(a)=0\) for every \(a \in A\).

(2)\(f(b) =1\) for every \(b \in B\).

(3) \(f(x) \le r\)r for any \(x \in \overline{U}_r\),

(4) \(f(x) \ge r\) for any \(x \notin U_r\).

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