CONTROLLING DISJOINTNESS WITH A HEDGEHOG

After Frantz's idea of controlling properties of extensions of continuous functions there has been an interest in extending families of continuous pairwise disjoint real-valued functions on normal spaces. We make the observation that disjoint extending a disjoint family of continuous functions is the same thing as extending a single continuous function with values in a hedgehog J(k) viewed as a bounded complete domain with its Lawson topology where k is the amount of pairwise disjoint functions which have to be extended. We provide a number of characterizations of spaces for which J(k) with its Lawson topology becomes an absolute extensor. In particular, this closes the circle of results related to disjoint extension theorems for normal spaces.