The Smirnov compactification as a quotient space of the Stone-Čech compactification (Q1112364)

The Smirnov compactification (X,\(\Delta)\) of a separated proximity space (X,\(\delta)\), with associated topology \(\tau =\tau (\delta)\), is obtained as a decomposition space of the Stone-Čech compactification \(\beta\) X of (X,\(\tau)\). Assume (X,\(\tau)\) to be embedded in \(\beta\) X as a subspace. An equivalence relation \(\sim\) on \(\beta\) X is defined as follows: \(p\sim q\) if and only if for all subsets A,B of X: if \(p\in cl A\) and \(q\in cl B\) then \(A\delta\) B. If \(\theta\) denotes the decomposition map \(\beta\) \(X\to \beta X/\sim\), then \(\sigma\) X can be idenfied with \(\theta\) (\(\beta\) X), and \(\theta\) \(A\Delta\) \(\theta\) B (in \(\sigma\) X) if and only if cl \(A\cap cl B\neq \emptyset\) (in \(\beta\) X). The proof is elementary, and provides a convenient base for obtaining the fundamental properties of the Smirnov compactification.