Continuity mistake: When at home, Selena says that the world is at her wish and looks down at the bronze sculpture which she is holding close to her chin. In the following angle her hand swaps to stomach level.
Continuity mistake: Supergirl's bracelet swaps wrists while she's flying at night.
Continuity mistake: When Selena is holding the glass champagne and says "I can't wait 'till the world is mine", from a back shot her finger is below liquid level, in the following shot it is above it.
Revealing mistake: When Selena uses her magical powers to accelerate the fairground ride to superfast speed, the twinkling fairground lights in the background are also flashing much faster. They obviously played the tape back at high speed to achieve this.
Continuity mistake: When Supergirl heats the monkey wrench, it swaps positions between shots.
Continuity mistake: The hole that Supergirl makes in the water tank changes shape between the close and wide shots.
Continuity mistake: In the girls' dorm, when Supergirl is trying the bra on, the teddy bear in the back changes position between shots.
Continuity mistake: Supergirl crushes a rock and gets dirt all over the sleeve of her suit. A second later dirt is gone and the suit is clean.
Continuity mistake: Selena starts to say that everything is history and turns around to say the world is hers. Right then, angle changes and cupboards appear on the right from nowhere. When the angle changes, the cupboards disappear.
Revealing mistake: When Supergirl is writing her letter of recommendation at super speed, it's way too obvious that the scene's been speeded up in several takes because the leaves in the background move fast and in a very weird and unnatural way.
Continuity mistake: When Supergirl enters her dorm, the Cornflakes box changes position from standing upright to sideways.
Continuity mistake: After her fight with the invisible monster, Supergirl walks by the broken steel door she fell against, and it looks more crumpled and worn out than before.
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.