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)

^{2}X = (CI)

^{4}X. 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 2

^{256}, which is way too big, but the topology axioms are fairly strong, aren't they?