Open mapping theorem

The Open Mapping Theorem states that if a function between two certain types of mathematical spaces (called Banach spaces) is continuous, linear, and surjective (meaning it covers all possible outputs), then it maps open sets in the first space to open sets in the second. In simpler terms, such a function preserves the "openness" of regions, so the image of an open area remains open. This result is important because it ensures that these functions don’t squash or distort space in a way that closes off regions, maintaining a certain level of "openness" under transformation.