A Note to Hutton's Theorem

Abstract:

      It is proved in this paper that there is a bijection $\varphi$ from ${\cal J}\cup{\cal J}'$ to $\delta\cup\delta'$ which satisfies: (1)~$\varphi|{\cal J}: ({\cal J},\subset)\longrightarrow(\delta,\leq)$ is a frame isomorphism; (2)~$\varphi|{\cal J}': ({\cal J}',\subset)\longrightarrow(\delta',\leq)$ is a coframe isomorphism, where ${\cal J}$ is the ordinary topology on $[0,1]$, $\delta$ is the ordinary $L$-topology on $L$-unit interval $I(L)$, and $L$ is a frame with an order-reversing involution. This result improves Theorem 3 in Hutton's paper.