The blog has become a little messy and I will try to clean it up. These simple implementations of sets and groups are limited but interesting. I believe in the idea of bundles on a stack, but the data stack is not perfect for the purpose: for the sake of portability the data stack can only be manipulated with a restricted number of words. With a special stack for bundles, stack manipulations can be made much more efficient.

While going on with the current implementation for a while, I will try to find out a faster implementation. With better algorithms: until now I have used sloppy straight forward brutal force algorithms or worse! Also the implementation of permutations and groups must be enhanced.

### Topological spaces

Topology is very easy to understand in spite of it's often formalistic presentation for the students. In set theory the question is: does an element belong to the set or not. In topology the question is if the element is close to the set or not.That's what a topological structure λ on a set X submit, the possibility to decide whether a point x in X belongs to the closure of a subset A of X or not. The set λ of open subsets of X brings the method: the point x is in the closure of A if and only if every open set O in λ that contain x, also contain a point in A.

Two subsets of X are obligatory in λ and that's ø and X. Beyond that there are two rules for λ to be a topology on X. (1) Given a subset B of λ, then the union of all sets in B should belong to λ. (2) given two sets O,Ô in λ, then the intersection of O and Ô should belong to λ.

__Examples__:

- The topology of the real numbers (a topology which is a necessary part of the definition of the set of real numbers) consists of all unions of limited open intervalls, that is, open intervalls of the type {x|a<x<b} for real numbers a,b. Any union of such intervalls belongs to this topology. The point 1 belongs to the closure of {x|0<x<1}, without actually being a member of that set, because every open set that contains 1 also contains a point in {x|0<x<1}.
- The set {ø,X} is a topology on X called the trivial topology and the set of all subsets of X is a topology called the discrete topology.
- Any infinite set X has an obvious topology consisting of all complements of finite subsets of X plus the empty set.
- Any set of subsets of a set X generates a minimal topology on X which contain those subsets.

In stack diagram for

**setcup**s is a set of sets and s' is a set of points. And s" is the set of the unions of all elements of s ans s':

{ { 1 2 } { 2 3 4 } } { 3 4 5 } setcup cr set.

{{1,2,3,4,5},{2,3,4,5}} ok

**: setcup ( s s' -- s")**

>>xst 0 >yst foreach

?do nxst@ union fence yst>> union >>yst

loop nxstdrop yst>> ;

>>xst 0 >yst foreach

?do nxst@ union fence yst>> union >>yst

loop nxstdrop yst>> ;

Similar for the word:

**: setcap ( s s' -- s")**

>>xst 0 >yst foreach

?do nxst@ intersection fence yst>> union >>yst

loop nxstdrop yst>> ;

>>xst 0 >yst foreach

?do nxst@ intersection fence yst>> union >>yst

loop nxstdrop yst>> ;

The word

**capgen**generates all intersections:

**: capgen ( s -- s')**

ndup >>yst ndup >>xst foreach

?do nyst@ nswap setcap xst>> union >>xst

loop nystdrop xst>> ;

ndup >>yst ndup >>xst foreach

?do nyst@ nswap setcap xst>> union >>xst

loop nystdrop xst>> ;

The word

**cupgen**generates the set of all unions os s. When first apply capgen and then cupgen all the non trivial open sets in the smallest topology that includes s are generated. And the word topology completes with ø and X, which might or might not be generated by

**capgen**and

**cupgen**.

**: cupgen ( s -- s' )**

ndup >>yst ndup >>xst foreach

?do nyst@ nswap setcup xst>> union >>xst

loop nystdrop xst>> ;

ndup >>yst ndup >>xst foreach

?do nyst@ nswap setcup xst>> union >>xst

loop nystdrop xst>> ;

{ { 1 2 } { 2 3 4 } { 3 4 5 } } capgen ndup cr set.

{{2},0,{3,4},{1,2},{2,3,4},{3,4,5}} ok

cupgen ndup cr set.

{{1,2,3,4},{1,2,3,4,5},{2,3,4,5},{2},0,{3,4},{1,2},{2,3,4},{3,4,5}} ok

{ 1 2 3 4 5 } { { 1 2 } { 2 3 4 } { 3 4 5 } } topology cr set.

{{1,2,3,4},{2,3,4,5},{2},{3,4},{1,2},{2,3,4},{3,4,5},{1,2,3,4,5},0} ok

**: topology \ X s -- λ**

nover nswap capgen

cupgen nswap incl 0 incl ;

nover nswap capgen

cupgen nswap incl 0 incl ;

In the figure the rectangle correspond to X and the colored shapes to the set of sets generating a topology. The colored surfaces are the primitive intersections, and the topology is the set of all combinations of unions of those intersections.

The next word computes the closure to the singleton {x}.

**: singlecl \ λ x -- s**

0 false loc{ x y flag } >>yst 0 >xst

nyst@ nunion foreach

?do to y true to flag nyst@ foreach

?do ndup y member

if x member 0=

if false to flag then

else ndrop

then

loop flag if xst>> y incl >>xst then

loop nystdrop xst>> ;

0 false loc{ x y flag } >>yst 0 >xst

nyst@ nunion foreach

?do to y true to flag nyst@ foreach

?do ndup y member

if x member 0=

if false to flag then

else ndrop

then

loop flag if xst>> y incl >>xst then

loop nystdrop xst>> ;

And the λ-closure for any set:

**: closure ( λ s -- s')**

nswap >>xst 0 >yst foreach

?do >r nxst@ r> singlecl yst>> union >>yst

loop nxstdrop yst>> ;

nswap >>xst 0 >yst foreach

?do >r nxst@ r> singlecl yst>> union >>yst

loop nxstdrop yst>> ;

{ 1 2 3 4 5 } { { 1 2 } { 2 3 4 } { 3 4 5 } } topology ok

{ 1 3 5 } closure cr set.

{1,4,3,5} ok

So except for the members 1,3,5 also 4 is close to {1,3,5}.

The word

**opsubs**computes all open sets that are subsets of s.

**: opsubs ( λ s -- s' )**

>>yst 0 >xst foreach

?do ndup nyst@ subset

if fence xst>> union >>xst

else ndrop

then

loop nystdrop xst>> ;

>>yst 0 >xst foreach

?do ndup nyst@ subset

if fence xst>> union >>xst

else ndrop

then

loop nystdrop xst>> ;

The interior of a set s is the set of all points which are members of an open set that is a subset of s. That is, s' is the union of all open subsets of s.

**: interior ( λ s --- s')**

opsubs nunion ;

opsubs nunion ;

{ 1 2 3 4 5 } { { 1 2 } { 2 3 4 } { 3 4 5 } } topology ndup ok

{ 1 3 5 } interior cr set.

0 ok

{ 2 3 } interior cr set.

{2} ok

{ 1 3 5 } interior cr set.

0 ok

{ 2 3 } interior cr set.

{2} ok

There is an error here. The topology isn't computed correct. Will fix.

ReplyDeleteNow I hope that everything is correct.

ReplyDelete