And if a named sub does, it's also a closure.

Indeed two unrelated concepts.

UPDATE

I just checked the etymology, the fact that the bound variables are not accessible from the outside of their defining scope makes the sub "closed".

{
    my $state = 42;

    sub closed_sub {
          return $state++;
    }
}

say closed_sub();  # 42
say closed_sub();  # 43
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Contrary to subs binding global variables, which are "open".

{
    our $state = 42;

    sub open_sub {
          return $state++;
    }
}

say open_sub();    # 42
$main::state = 666;
say open_sub();  # 666
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(of course Perl allows you to bind both kinds, so it's not an exclusive attribute here)

see references from WP: