In Chapter 13 "Atomic Objects" of the book "Distributed Algorithms" by Nancy Lynch, linearizability (also known as atomicity) is proved to be a safety property. That is to say, its corresponding trace property is nonempty, prefix-closed, and limit-closed, as defined in Section 8.5.3. Informally, a safety property is often interpreted as saying that some particular "bad" thing never happens.
Based on this, my first problem is as follows:
What are the advantages of linearizability as a safety property? Are there some results based on this fact in the literature?
In the study of the classification of safety property and liveness property, it is well-known that safety property can be characterized as the closed set in an appropriate topology. In the paper "The Safety-Progress Classification" @1993 by Amir Pnueli et al., a metric topology is adopted. More specifically, a property $\Phi$ is a set of (finite or infinite) words over the alphabet $\Sigma$. The property $A(\Phi)$ consists of all infinite words $\sigma$ such that all prefixes of $\sigma$ belong to $\Phi$. For example, if $\Phi = a^{+}b^{\ast}$, then $A(\Phi) = a^{\omega} + a^{+}b^{\omega}$. An infinitary property $\Pi$ is defined to be a safety property if $\Pi = A(\Phi)$ for some finitary property $\Phi$. The metric $d(\sigma, \sigma')$ between infinite words $\sigma$ and $\sigma'$ is defined to be 0 if they are identical, and $d(\sigma, \sigma') = 2^{-j}$ otherwise, where $j$ is the length of the longest common prefix on which they agree. With this metric, the safety property can be characterized as closed sets topologically.
Here comes my second problem:
How to characterize linearizablity as closed sets topologically? In particular, what is the underlying set and what is the topology?