How do you unlock the cave near lilycove city in emerald?

To unlock the cave near Lilycove City in Pokémon Emerald, you need to obtain the Surf HM and use it to navigate to the cave entrance, which is located on the water south of Lilycove. Once you reach the entrance, you'll find it leads to the Safari Zone. In order to access the cave itself, you must also have completed the necessary storyline involving Team Aqua and Team Magma, allowing you to progress in the game.