Mayonnaise can in fact be left out but you will want to make sure not to keep it somewhere that is too warm because of the fact that it can get oily over time which makes it quite unappetizing. It is safe to leave mayonnaise outside of the refrigerator though, even after you have opened it. Because there are so many different additives put in which act as preservatives you should be able to leave your mayo out for a long time before there is a risk of it spoiling.
You will however want to make sure to take a look at the expiration date. Some people tend t leave food and condiments in their refrigerator longer than they should. But mayonnaise has a very long period of time where it is good to eat, so unless you keep it sitting out for a very long time you shouldn’t have any problems with bacteria growing.