Get names of all keys in the collection

I’d like to get the names of all the keys in a MongoDB collection.

For example, from this:

db.things.insert( { type : ['dog', 'cat'] } );
db.things.insert( { egg : ['cat'] } );
db.things.insert( { type : [] } );
db.things.insert( { hello : []  } );

I’d like to get the unique keys:

type, egg, hello