10. HIERARCHY OF SINGULARITIES 23
Moreover, if d is T
2
-separable, they are also isomorphic to the case
X = d ◦ d
−1
(remark: for a pretopology d, it’s the proximity of two
sets being near if they have intersecting closures). The isomorphism is
composing every element F of an axiomatic singularity with d on the
left (F 7→ d ◦ F ).
Also let question how to generalize the above for functions φ be-
tween diﬀerent kinds of singularities is also not yet settled.
10. Hierarchy of singularities
Above we have deﬁned (having ﬁxed endofuncoids µ and X) for
every set of “points” R = Ob X its set of singularities SNG(R).
We can further consider
SNG(SNG(R)), SNG(SNG(SNG(R))),
etc.
If we try to put our generalized derivative into say the diﬀerential
equation h◦f
0
= g◦f on real numbers, we have a trouble: The left part
belongs to the set of functions to SNG(Y ) and the right part to the set
of functions to Y , where Y is the set of solutions. How to equate them?
If Y would be just R we would take the left part of the type SNG(R)
and equate them using the injection τ deﬁned above. But stop, it does
not work: if the left part is of SNG(R) then the right part, too. So the
left part would be SNG(SNG(R)), etc. inﬁnitely.
So we need to consider the entire set (supersingularities)
SUPER(R) = R ∪ SNG(R) ∪ SNG(SNG(R)) ∪ . . .
But what is the limit (and derivative) on this set? And how to
perform addition, subtraction, multiplication, division, etc. on this set?
Finitary functions on the set SUPER(R) are easy: just apply τ to
arguments belonging to “lower” parts of the hierarchy of singularities a
ﬁnite number of times, to make them to belong to the same singularity
level (the biggest singularity level of all arguments).
Instead of generalized limit, we will use “regular” limit but on the
set SUPER(R) (which below we will make into a funcoid) rather than
on the set R.
See? We have a deﬁnition of (ﬁnite) diﬀerential equations (even
partial diﬀerential equations) for discontinuous functions. It is just a
diﬀerential equation on the ring SUPER(R) (if R is a ring).
What nondiﬀerentiable solutions of such equations do look like? No
idea! Do they contain singularities of higher levels of the above hierar-
chy? What about singularities in our sense at the center of a blackhole