The Tamano-Dowker type theorems for nearness frames (Q1892301)

A nearness frame \((L, \mu)\) is a frame \(L\) together with a compatible nearness \(\mu\) on \(L\). This means \(L\) is a regular frame. In this paper the author introduces notions of regularity and normality for nearness frames, defines the coproduct of two nearness frames and gives necessary and sufficient conditions in order that a nearness frame \((L, \mu)\) is normal. Let \((o[0,1], \text{Cov} (o[0,1]))\) be the nearness frame with the frame \(o[0,1]\) of all open sets of the unit interval \([0,1]\) and the nearness \(\text{Cov} (o[0,1])\) consisting of all covers of \(o[0,1]\). Then \((L, \mu)\) is normal iff the coproduct of \((L, \mu)\) and \((o[0,1], \text{Cov} (o[0,1]))\) is normal. The author gives other equivalent conditions expressed by the normality of the coproduct of \((L, \mu)\) and \((M, \nu)\) for some nearness frame \((M, \nu)\). These results are the localic version of the Tamano-Dowker type theorems for nearness spaces as obtained by \textit{H. L. Bentley}. The proofs are accomplished in the realm of the frame theory without using points. As an application it is shown that for a Dowker space \(X\) the coproduct of \((oX, \text{Cov} (oX))\) and \((o[0,1], \text{Cov} (o[0,1]))\) is a normal nearness frame, while the coproduct of \(oX\) and \(o[0,1]\) is not a normal frame.