How can i show that the empty subset of a metric space,X always connected ?

It is empty and so will have and so will have an empty boundary.That doesn't seem to be enough.

Also, my book says that, no other finite set can be connected,I don't really understand this, because every finite set will be a non empty proper subset of X and will always be closed (and never both open and closed) so shouldn't it be connected ?

Any help will be appreciated.