Another approach to extensions of continuous mappings

First, we give a more general extension theorem about continuous mappings. It is shown that the Taimanov-Eilenberg-Steenrod extension theorem and the Engelking extension theorem are special cases and that this theorem implies the Dugunji extension theorem. The localic version of this theorem also generalises Joyal's extension theorem for locales. Then, using the same technique, we obtain another more interesting extension theorem and its applications. In particular, we sharpen a well-known result due to Wallman.