Addenda and errata for Natural Topology (2nd ed.)

[Never my finest moments: discovering flaws in what I tried so hard to create as a perfect piece of work…:-)]

I’m rereading the second edition of Natural Topology (available at http://www.fwaaldijk.nl/natural-topology.pdf, as well as on arXiv), and naturally I am spotting some omissions, typos and even errors.

I will make a list of these in this post, and I will update this list until I replace the second edition with a third (which then in turn I fear will still be in need of a similar list, but one hopes for improvement along the line).

The most important change to be made is related to theorem 1.2.2. The given proof in A.3.1 of this (beautiful) theorem is partly deficient, because it fails to take into account the strict requirement (i) in definition 1.1.2. of `morphism’. At the time I saw no need to relax this requirement, since everything seemed to work smoothly, and in all `regular’ situations this requirement is fulfilled. So I opted for some form of aesthetic optimality.

In hindsight, I should have noted that the requirement (i), which is phrased for dots, is too restrictive in a pointwise setting. Thankfully the remedy is easy: simply replace this with the slightly less restrictive pointwise phrasing (see below), and all is well. No need even to change any other wordings, in proofs or elsewhere.

But in almost all relevant situations, the requirement (i) is easily met. I would like to mention this in an elegant way, without making a separate distinctive definition of say `morfism’ (to be pondered on). The other addenda and errata are all very minor, so far.

Errata:

(X) Definition 1.1.2. should read:

“Let $(\mathcal{V},\mathcal{T}_{\#_1})$ and $(\mathcal{W},\mathcal{T}_{\#_2})$ be two natural spaces, with corresponding pre-natural spaces $(V,\#_1, \preceq_1)$ and $(W,\#_2, \preceq_2)$ . Let $f$ be a function from $V$ to $W$ Then $f$ is called a refinement morphism (notation: $\preceq$ -morphism) from $(\mathcal{V},\mathcal{T}_{\#_1})$ to $(\mathcal{W},\mathcal{T}_{\#_2})$ iff for all $a,b\in V$ and all $p=p_0,p_1,\ldots,\ q=q_0,q_1,\ldots\in\mathcal{V}$ :

(i) $f(p)=_{\rm D}\ f(p_0), f(p_1), \ldots$ is in $\mathcal{W}$ (`points go to points’)

(ii) $f(p)\#_2 f(q)$ implies $p\#_1 q$ .

(iii) $a\preceq_1 b$ implies $f(a)\preceq_2 f(b)$ (this is an immediate consequence of (i))

As indicated in (i) above we will write $f$ also for the induced function from $\mathcal{V}$ to $\mathcal{W}$ . The reader may check that (iii) follows from (i). By (i), a $\preceq$ -morphism $f$ from $(\mathcal{V},\mathcal{T}_{\#_1})$ to $(\mathcal{W},\mathcal{T}_{\#_2})$ respects the apartness/equivalence relations on points, but not necessarily on dots since $f(a)\#_2 f(b)$ does not necessarily imply $a\#_1 b$ for $a,b\in V$ . This stronger condition however in practice obtains very frequently.”

Addenda:

(O)(in the terminology of erratum (X) above):

If $f(a)\#_2 f(b)$ , then by (ii) we know that $x\#_1 y$ for all $x\prec a, y\prec b$ in $\mathcal{V}$ . Therefore, if necessary we could `update’ the apartness to ensure $a\#_1 b$ …but we cannot guarantee that this is simultaneously possible for all similar pairs of dots $c,d$ in $V$ .

However, most spaces $(\mathcal{V},\mathcal{T}_{\#})$ naturally carry an apartness on dots such that if $x\# y$ for all $x\prec a, y\prec b$ in $\mathcal{V}$ , then $a\# b$ . In this situation, (ii) of the definition becomes equivalent with (ii’): $f(a)\#_2 f(b)$ implies $a\#_1 b$ for all $a,b\in V$ . (This (ii’) is part of the original definition 1.2.2., which should be replaced by the above definition.)

(OO) It should be noted that $(\mathcal{V}^{\wr\wr}, \mathcal{T}_{\#})$ is $\preceq$ -isomorphic to $(\mathcal{V}^{\wr}, \mathcal{T}_{\#})$ . This means that it always suffices to look at $\mathcal{V}^{\wr}$ .

(OOO) It should be noted that all natural spaces ‘are’ spreads already, when looking at their set of points. This is another (perhaps easier) way of seeing that any natural space is spreadlike. Let $(\mathcal{V},\mathcal{T})$ be a natural space with corresponding pre-natural spaces $(V,\#, \preceq)$ . Assume $h$ is an enumeration of $\{(a,b)\subset V\times V\mid a\# b\}$ . To create a point $x=x_0, x_1, ...$ in $\mathcal{V}$ , one can start with any basic dot $a$ as $x_0$ . Then, one chooses $m_0\in N$ , and for the next $m_0$ values of $x$ one is free to choose basic dots $x_{(m_0)}\preceq...\preceq x_0$ , but at stage $m_0+1$ we must choose for $x_{(m_0+1)}$ a basic dot $c$ such that $c$ is apart from at least one of the constituents of $h(0)$ . Then one chooses $m_1\in N$ , etc. Therefore we see that, if one disregards the partial order, we did not create any new structure outside of Baire space. And there is no problem whether our point are sets (contrasting to formal topology, where there seems to be a problem in general whether the points of a formal space form a set).