I have a question. I read many blogs, websites about CAP theorem. They say that single-node systems are CA, but how can it be "A" if that single node goes down ? Because if it does, the system will be unavailable, right ? And is "availability" in CAP equal to "availability" in HA of distributed systems ?
How does a single-node system get Availability in CAP theorem?
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That is not the definition of (cap) availability, it is not about being able to reply but refusing to do so, e.g. the node receives a message and it is not able to contact the rest of the cluster, what can it do: i) respond thereby becoming A but not C; ii) do not respond and become C but not A. See my previous answer here for illustrative examples.
As for your specific example I'm not sure it makes sense to reason about CAP if the cluster does not exist, I believe that you need at least one node.