The Arbor Day organization say 10 feet apart for dwarf
apple trees. I have heard of trees in intensive plantings as close as 4 to 5 feet apart. You should also take into account your soil type, in a heavy soil
roots will stay closer to the surface so you should give them more space, in sandy soil the the roots will go deeper an compete less. If you want the trees closer together consider planting on a
berm or
hugel bed, planting trees above the soil level will encourage the roots to go down.