The convexity induced by quasi-consistency and quasi-adjacency

Abstract

In this paper, we introduce (quasi-)consistent spaces and (quasi-)adjacent spaces to characterize convexity spaces. Firstly, we show that convexity spaces can be characterized by quasi-consistent spaces. They can be induced by each other. In particular, each convexity space can be quasi-consistentizable. Every quasi-consistency $\mathcal{U}$ can induce two hull operators and thus determine different convexities $\mathcal{C}^{\mathcal{U}}$ and $\mathcal{C}_{\mathcal{U}}$. And $\mathcal{C}^{\mathcal{U}}=\mathcal{C}_{\mathcal{U}}$ holds when $\mathcal{U}$ is a consistency. Secondly, we use quasi-adjacent spaces to characterize convexity spaces. Each convexity space can be quasi-adjacentizable. In both of characterizations of convexity, remotehood systems play an important role in inducing convexity. Finally, we show there exists a close relation between a quasi-consistency and a quasi-adjacency. Furthermore, there exists a one-to-one correspondence between a quasi-adjacency and a fully ordered quasi-consistency. And we deeply study the relationships among these structures.

Keywords

• convexity
• remotehood system
• quasi-consistency
• quasi-adjacency

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