Addendum to Natural Topology regarding functions

In the appendix, I omitted to include the definition of `function´ in section A.4 Constructive concepts and axioms used. The reason is that we use the standard classical definition:

Having taken the notion of `set´ and `subset´ as primitive, we define a function f from an apartness space (V,\#_1) to another apartness space (W,\#_2) as a  subset of the cartesian product V\times W such that:

i) for all x\in V there is a y\in W such that (x,y)\in f

ii) for all x,v\in V, y,z\in W: if (x,y)\in f and (v,z)\in f and y \#_2 z then x \#_1 v.

Then for any pair (x,y)\in f we write: f(x)\equiv y or f(x)=y.


The constructive interpretation of the quantifiers `for all´ and `there is´ ensures in our eyes that this definition nicely captures the connotation of methodicity which the word `function´ carries. In the book we almost always work with morphisms anyway, but the definition above is strictly speaking necessary to underpin the theorems on representability of continuous functions by morphisms.



About fwaaldijk

mathematician (foundations & topology in constructive mathematics) and visual artist
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