FOM: 5:Constructions

[Vaughan R. Pratt]

>The most common situation is illustrated by, e.g., the
>"construction" of the ordering of the rationals as "the countable dense
>linear ordering without endpoints," which "defines" the ordering of the
>rationals uniquely up to isomorphism.

Some hard-to-please people might grumble that being a set that is order
isomorphic to the rationals is not much of a number system without any
arithmetic operations defined on it.  

Would the following example serve the same purpose?  Define the
continuous functions to be the largest class of functions between
topological spaces such that

(i) it is closed under composition, and

(ii) every continuous function to the Sierpinksi space ({0,1} with {1}
open but not closed) is the characteristic function of an open set.

The advantage of this example is that one really can imagine a topology
instructor wanting to use this definition to make continuity seem less
a contrived notion.

The reason I don't use this definition myself is precisely because of
the problem you raise.  In this case the topological spaces themselves
already form a class, as hence do the functions between them, and
there's no wriggling out of your problem by finding a suitably large
set to contain them all.

Vaughan