Are there any rules against making a catch with the hand without a mitt? Will the batter still be out?
Can a baseball be caught with the non-mitt (no glove) hand?
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are there any rules against taking a catch with the hand without the Mitt on it?
There are no rules against catching a ball in flight with the hand. See the definition for "catch" below.
Will the batter still be given out?
Yes. See Rule 5.09 below.
The definition for "catch" can be found in the 2018 MLB Official Baseball Rules (emphasis mine):
A CATCH is the act of a fielder in getting secure possession in his hand or glove of a ball in flight and firmly holding it; providing he does not use his cap, protector, pocket or any other part of his uniform in getting possession. It is not a catch, however, if simultaneously or immediately following his contact with the ball, he collides with a player, or with a wall, or if he falls down, and as a result of such collision or falling, drops the ball. It is not a catch if a fielder touches a fly ball which then hits a member of the offensive team or an umpire and then is caught by another defensive player. In establishing the validity of the catch, the fielder shall hold the ball long enough to prove that he has complete control of the ball and that his release of the ball is voluntary and intentional. If the fielder has made the catch and drops the ball while in the act of making a throw following the catch, the ball shall be adjudged to have been caught.
Rule 5.09 states:
A batter is out when -
(1) His fair or foul fly ball (other than a foul tip) is legally caught by a fielder
not only is it legal, it's been done!
Here's the proof, from 1989, by the San Francisco Giants' Kevin Mitchell: