[FOM] constructive continuity 2

Frank Waaldijk frank.waaldijk at hetnet.nl
Thu Feb 7 20:55:32 EST 2008

(For the moderator: this message is posted here because the issues below 
seem to me of importance to the foundations of constructive mathematics, and 
I would like to draw attention and reactions to these issues.)

Formal topology is quoted in several places to be a constructive way of 
doing topology, in which the absence of the fan theorem can be compensated 
by the use of inductively defined covering relations on the basic opens of a 
formal space. It is, to freely quote Giovanni Sambin, a quickly developing 
field, in which several different approaches are tried out.

On rereading Erik Palmgren's article `Continuity on the real line and in 
formal spaces' (see http://www.math.uu.se/~palmgren/crealform-rvd2.pdf), I 
find it difficult to reconcile its results with a theorem in my paper `On 
the foundations of constructive mathematics-especially in relation to the 
theory of continuous functions'.

Since the basic definitions regarding formal topology seem to vary, I think 
it not unlikely that this is a definitional matter. Perhaps someone 
well-versed in formal topology could help me understand where the trouble 
lies? This might help in improving my understanding and/or the 

The reason for this question is (as in my first message) that the results of 
Palmgren's paper are quoted both in the paper itself and in Peter Schuster's 
paper `What is continuity, constructively?' to be a solution to the problems 
raised in my paper above.

In Erik Palmgren's paper:

Theorem 4.1 states that every Bishop-continuous function from R to R is 
representable by a continuous mapping A_f from {R} to {R} (where I use {R} 
to denote the formal real line)

Theorem 4.6 states in effect the converse: every continuous mapping from {R} 
to {R} represents a Bishop-continuous function from R to R.

Lemma 5.2 is used to show that the reciprocal function x-->1/x is 
representable by a continuous mapping, say Z.

Proposition 3.3 is used in stating that the composition of continuous 
mappings is a continuous mapping.

Now my problem:
Surely Thm 4.1 means that every Bishop-continuous function from [0,1] to R 
is representable by a continuous mapping?

But Corollary 4.4 of my paper above (which I refer to as [Waaldijk2005]) 

In BISH the following statements are equivalent:
1) The fan theorem
2) If f is a Bishop-continuous function from [0,1] to R+, then the 
composition 1/f of  f with the function x-->1/x is again Bishop-continuous

Combining the results we find (?)

If f is a Bishop-continuous function from [0,1] to R+, then A_f is a 
continuous mapping from the formal interval {[0,1]} to {R}+ (Thm 4.1 
Palmgren). Then the composition of A_f with the reciprocal mapping Z 
(representing x-->1/x) is again a continuous mapping (by Lemma 5.2 and Prp. 
3.3 Palmgren), say W from {[0,1]} to {R}+. Then W represents the function 
1/f which (by Thm 4.6 Palmgren) is Bishop-continuous.

Therefore (by Crl. 4.4 Waaldijk) the fan theorem holds.

I cannot pinpoint the reason for this unexpected derivation of the fan 
theorem, but I suspect that it lies somewhere in the definitions involved in 
creating the mapping Z representing the reciprocal function x-->1/x. Or 
perhaps there is some unforeseen difficulty in moving from {R} to the formal 
interval {[0,1]}? In both cases this seems to me to warrant some further 

All elucidation and likely correction of some oversight on my part are 

Frank Waaldijk


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