I woke up this morning and realized Kuratowski's "14" theorem isn't that hard to prove after all.

The theorem says that if you start with some subset of a topological space and your only tools are taking the complement of a set, and taking the closure of a set, then the maximum number of different sets you can generate (including the first one) is 14. This struck me as a monstrous result when I was a wee little undergrad --- what business does such a weird number as 14 have, intruding so upon the stage of legitimate mathematics! It should know its rightful place --- successor of 13, predecessor of 15 --- and keep to it. The real business of mathematics is reserved for nice, round numbers like 1024 and such.

In spite of that, the answer is really 14. A way to see that it's at most 14 is to write closure as C, complement as I, and put N = ICI for the interior of a set. We have by the definition of a topology and basic set theory that

X ⊆ CX
X ⊆ Y implies CX ⊆ CY
CCX = CX
X ⊆ Y implies IY ⊆ IX
IIX = X

so it's very easy to check that

NX ⊆ X
X ⊆ Y implies NX ⊆ NY
NNX = NX

and consequently

CNX = CNNX ⊆ CNCNX
CNCNX ⊆ CCNX = CNX

hence CNX = CNCNX, and expanding definitions this means (CI)2X = (CI)4X. We can never get anywhere by hitting a set with C or I twice in a row, so the only useful sequences are alternating sequences of C and I. Thus the only ones we can get are
X
CX
ICX
CICX
ICICX
CICICX
ICICICX
(but CICICICX = CICICICIIX = CICIIX = CICX)
IX
CIX
ICIX
CICIX
ICICIX
CICICIX
ICICICIX
(but CICICICIX = CICIX)

Count 'em up, it's 14. To prove that these are all obtainable in some topological space (i.e., to prove that it's at least 14), consider the topological space

X = (**)(*(*))(*(*)*)((*)*)

(where the *s are the 9 points, and the basic open sets are given by parenthesized ranges)

and the subset Y ⊆ X given by the @-signs in

(@*)(@(*))(@(*)*)((@)*)

Then we get
``` ICICICY = (**)(@(@))(*(*)@)((*)*)
CICICY = (@@)(*(*))(@(@)*)((@)@)
ICICY = (@@)(*(*))(@(*)*)((@)@)
CICY = (**)(@(@))(*(@)@)((*)*)
ICY = (**)(*(@))(*(*)@)((*)*)
CY = (@@)(@(*))(@(@)*)((@)@)
Y = (@*)(@(*))(@(*)*)((@)*)
IY = (*@)(*(@))(*(@)@)((*)@)
CIY = (@@)(@(@))(*(@)@)((*)@)
ICIY = (**)(*(*))(@(*)*)((@)*)
CICIY = (**)(*(*))(@(@)*)((@)@)
ICICIY = (@@)(@(@))(*(*)@)((*)*)
CICICIY = (@@)(@(@))(*(@)@)((*)*)
ICICICIY = (**)(*(*))(@(*)*)((@)@)
```

I think computationally finding the minimal topology that has this property might just almost be tractable. Maybe I'm vastly underestimating the number of topologies on n elements... The double powerset is 2256, which is way too big, but the topology axioms are fairly strong, aren't they?
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