Why is linearizability a safety property and why are safety properties closed sets?
What are the advantages of linearizability as a safety property? Are there some results based on this fact in the literature?


Suppose that you've implemented a shared memory machine $M$ that only satisfies eventual linearization, defined as follows: in every run $\alpha$ of $M$, there exists some point in time $T_\alpha$, such that linearization holds from time $T_\alpha$ on. Note that there is no upper bound on $T$. (*)
 (This is an artificial liveness counterpart of the standard safety property definition of linearizability.)

Such a shared memory implementation wouldn't be very useful to the programmer: Note that if only eventual linearizability holds, there are no guarantees whatsoever on the consistency of read/write operations in any "early" prefix of a run (before the unknown time $T$). Or, in other words, whatever has happened until now, you can still extend the current prefix of a run to one that satisfies eventual linearizability.  

(*) If there was such an upper bound, then eventual linearizability would become a safety property. 


  How to characterize linearizablity as closed sets topologically? In particular, what is the underlying set and what is the topology?


We can define a metric topology on the set $ASYNC$, which is the set of all possible runs of a distributed algorithms. Note that each run $\alpha \in ASYNC$ corresponds to an infinite sequence of state transitions. 
For $\alpha, \beta \in ASYNC$, $\alpha \ne \beta$, we define $$d(\alpha,\beta) := 2^{-N}$$where $N$ is the earliest index where the state transitions in $\alpha$ and $\beta$ differ; otherwise, if $\alpha = \beta$, we define $d(\alpha,\beta) = 0$.

We first argue that $d$ is a metric on $ASYNC$.
By definition, $d$ is nonnegative and $\forall \alpha,\beta \in ASYNC$ we have
$d(\alpha,\beta)=d(\beta,\alpha)$.
For $\alpha,\beta,\gamma \in ASYNC$, the triangle-inequality $d(\alpha,\beta)
\le d(\alpha,\gamma) + d(\gamma,\beta)$ trivially holds if $\gamma=\alpha$ or
$\gamma=\beta$.
Now consider the case that $d(\alpha,\gamma) \ge d(\gamma,\beta) &gt; 0$, i.e.,
$d(\alpha,\gamma)=2^{-n_1}$ and $d(\gamma,\beta)=2^{-n_2}$, for some indices
$n_1\le n_2$.
Since $\gamma$ shares a common prefix of length $n_2-1$ with $\beta$ but only a
prefix of length $n_1-1$ with $\alpha$, it follows that $\alpha$ and
$\beta$ differ at index $n_1$, and thus $d(\alpha,\beta) = d(\alpha,\gamma)$ and
the triangle-inequality follows.
The case where $0&lt;d(\alpha,\gamma) &lt; d(\gamma,\beta)$ follows analogously.

The metric $d$ induces a topology (e.g., page 119 of [1]) where the $\epsilon$-balls  $B_\varepsilon(\alpha) = \{
\beta \in ASYNC \mid d(\alpha,\beta) &lt; \varepsilon \}$ are the basic open sets.
We will now argue why safety properties correspond to
closed sets:
If an execution $\alpha$ does not satisfy a safety property
  $S\subseteq ASYNC$, i.e.\ $\alpha \notin S$, then there is an index 
  $N$ where
  all runs $\beta$ that share a prefix longer than $N$ with $\alpha$ are
  not in $S$. This closely matches intuition, since once a 
    safety property is violated in a prefix of an execution, it makes no 
    difference how this prefix is extended!
Formally speaking, suppose that $\alpha \notin S$. 
There exists an $N\geq 0$ such that, if some
$\beta \in ASYNC$ has $d(\alpha,\beta) &lt; {2^{-N}}\text{,}$ i.e., $\alpha$ and 
$\beta$ share a prefix of length $\ge N$, then $\beta \notin S$.  Thus, the 
set of runs $S$ is closed, since its complement is open.

[1] James Munkres. Topology.