Dawson–Gärtner theorem

The Dawson–Gärtner theorem is a result in the field of topology that deals with the relationship between compact spaces and their continuous images. It specifically addresses the conditions under which a continuous image of a compact space is also compact. The theorem states that if \(X\) is a compact space and \(f : X \to Y\) is a continuous function, then the image \(f(X)\) is compact in \(Y\).