*Defn 1:* a **go-space** (X,T,S) is a topological space (X,T) together with a subset
S of T which does not contain the empty set.

__Rule 1:__ Play is on a go-space (X,T,S) by two players called Black and
White.

__Rule 2:__ Each element of X may be coloured black, white or empty.

*Defn 2:* If an element of S, (s say) is such that all elements of s are
coloured a single colour then s is said to be **coloured** that colour (So
an element of S is either black, white, empty, or non-coloured - i.e.
mixed).

*Defn 3:* A **stone** is an element of S which is coloured black or white.

*Defn 4:* A **legal placement** is an element of S which is coloured empty.

*Defn 5:* A set of stones all of which have the same colour is **go-connected** if the closure of the union
of the stones is T-connected.

*Defn 6:* A go-connected set of stones G has **liberty** if there exists a
legal placement e, such that if e were coloured with the same colour as the stones of G then {e} union G would be go-connected.

*Defn 7:* the **string** of a stone s, or **string(s)**, of
colour C is the union over all subsets A of {t in S : t is of colour C} such that A contains s, and
A is go-connected.

*Defn 8:* **Clearing a colour** C is the process of colouring empty all elements
in:

{x in X such that x is of colour C, and for all stones s of colour C in S : x is in s, string(s) does not have liberty}

*Defn 9:* A **board play** consists of choosing a legal placement e, colouring
all elements of e one's own colour; then clearing the opponent colour, and
then clearing one's own colour.

__Rule 3:__ The game starts with all elements of X coloured empty, the players
alternate turns (a,b), starting with Black.

### Primitive Rules

*Defn 10a:* A **turn** is a board play that doesn't repeat an earlier X colouring.

__Rule 4a:__ The first player unable to take their turn is the loser.

### Cardinality 'Stone' Scoring Rules

*Defn 10b:* A **turn** is either a pass or a board play that doesn't repeat an earlier X colouring.

*Defn 11b:* A player's **score** is: |{x in X : x is of that player's colour}|

__Rule 4b:__ The game ends after 2 consecutive passes.

__Rule 5b:__ The player with the higher score (in terms of cardinality) at the end of the game is the winner. Equal scores result in a tie.

### Acknowledgements

This is derived from the Tromp-Taylor rules, scoring from primitive rules.

## Comments

**D1** - S is the set of possible stone placements. S is totally general as
long as it doesn't contain the empty set, though I'm trying to work out general principles of choosing (X,T,S)
to get a reasonable game (e.g. one in which it's actually possible to put a stone down...). The empty set not being in
S is essential unless we are going to put 'non-empty' conditions all over the place. For example, if the empty set is in S with
the other definitions as they are now, its impossible to take anything since we can take e the empty set in Defn 6.

**R2** - We could define colouring more rigorously as a
function from X to {b, w, e}, then D2 as another function defined from this one. However, there's still an element of concepts 'outside of the maths'
with players then altering this function as play goes on. Perhaps I'd need to take a look
at some game theory in order to work out how to formalise taking turns (and altering the
colouring function as 'time' goes on) in a convenient way.

**D2** - Note that I'm using the word 'colour' in two different ways, one for colour of elements
of X, and one as defined here for colour of sets in S. It should be clear which one I mean, and in the definitions the links
take you to the definition of the relevant use of the word 'colour' (R2 or D2).

**D3** - If I defined the stones to be the elements of X then I think it would reduce
to general graph go however you define connection.

**D4** - Note with this and D3 I'm just giving names to coloured elements of S referred
to in D2 for convenience later. Interestingly, the definitions for the concepts of stone or legal placement are nearly
identical, though the uses are very different.

**D5** - This is nice - you don't need a separate idea of connecting by moving
horizontally and vertically along grid lines - the topological connectedness
property does it for you. I've also considered saying 2 stones are
go-connected if the intersection of their closures is non-empty. This
doesn't have that nice property, but might be simpler to work with in some cases.

One of the topological definitions of a set U being T-connected (the one I'm using) is if within the induced topology (U,TU) there are no clopen sets other than U and the empty set. TU is the set {U intersect t : t is in T}, a clopen set is a set which is open and closed.

Previously I took the union of the closures of the stones, but this isn't the same as the closure of the union in general infinite cases. The closure of the union is slightly 'bigger'.

**D6** - We don't define liberties (plural), since its not...well defined here.
You can have situations where you can fit say, 2 legal placements next to
a stone, or if you do it differently then 3 legal placements.

**D7** - This definition can deal with any amount of infinite strings, boards etc.
with no problems - it only uses set theoretic concepts. Previously I defined this with a 'maximal' go-connected
subset, which is somewhat unrigorous. There is a proof below that string(s) is go-connected.

**D8** - I think its better to define the colouring on X rather than on S, in
order to allow a little more generality. If S contains non-connected sets
then they generally die as soon as they are placed, unless there happens
to be a 'liberty placement' which touches all parts of the stone. With
this definition you can place the 'stone' and each part of it can act as
a separate entity if there happens to be an element of S which works like
one of the parts of the played stone. Basically it allows topological go
to encompass variants in which you play more than one stone at a time.
We don't get any weird effects such as bits of stone being left on the board whilst not
being capturable or otherwise acting like normal stones, because of this result.

**D10a,b** - In effect, D10a is **primitive ko**, and D10b is
**positional super ko**

**R4a,b** - The game might never end. Even if we make the players take each
move in half the time of the previous move, if there are an uncountable
number of elements in S then it's possible for the game to still have an
uncountable number of moves left. The problem with uncountable numbers
of moves is that black and white are supposed to alternate and effectively
list their moves, but then the uncountable would be countable. Maybe there's
a way round somehow...axiom of choice?

**Scoring** - Cardinality 'stone' scoring will give a draw in many games with X infinite. Primitive rules
do not, but the game is somewhat different from most regular go rule sets when we specialise to
normal 2d go.

## Examples

### 1) Regular go with primitive rules

If we take X to be [0.5,19.5]x[0.5,19.5], T to be the usual Euclidean topology (well, the induced topology on X), S to be sets of the form:{(x,y) : (x-p)^2+(y-q)^2<1/4, for p, q, integers in [0,19]}

...then we get the usual 2 dimensional game of go with primitive rules (no passes). Cardinality 'stone' scoring doesn't work (it's well defined but will not distinguish a winner) since if a player has a single live stone at the end of the game they have score uncountably infinite.

### 2) General graph go

...is a specialisation of Topological go in the following way (though there are other ways):

If the graph is a set of vertices V, with a set of edges E, then define:

X = {xv : v is in V} union {yuv : an edge joining u and v is in E}

...where the xv and yuv are all distinct. (note the terminology for yuv gives yuv=yvu)

T has a base:

{{xv} : v is in V} union {{xv, yuv, xu} : u,v are in V and joined by an edge}

S is the sets:

{xv} where v is in V

With these definitions we should have go on a general graph. I may get around to proving this statement fully, using this definition of general graph go - see below.

With a general graph we can of course get another go space which acts like regular go, though we can easily specify to diamond structure instead.

## Some Results

### 1) The specialisation to General Graph go

For now just a check to see if adjacient stones are go-connected, whereas non-adjacient stones are not:

Suppose xu, xv are in X as above, and u is adjacient to v (i.e. yvu is in X). First we need to work out the closure of {xu}. This is defined to be the intersection of the closed sets containing {xv}. Equivalently p is in the closure of xv if p is not in any open set not containing xv.

The elements in X satifying this are xv itself, and yvw, for w some vertex. Similarly for xu. So the union of the closures is U := {xv, xu, yuv} union {yvw, yuw : w is in V, if such y's exist in X}.

*Claim:* This set is T-connected.

*Proof:*
We show that there are no clopen sets in U other than the empty set and U itself. Consider the element
yuv. We wish this element to be in an open set - but any open set which contains
yuv must also contain xu and xv.
If there are no other elements in U then we are done. Suppose (without loss of generality) there exists
a yuw in U, where w =/= v. Suppose for contradiction we can find an open set containing
yuw other than the set containing yuv. In the induced topology on U,
if yuw is in an open set, then xu is also in that set. So yuw
must be in the set also containing yuv. Contradiction. So there are no clopen sets in U other than
U and the empty set, so U is connected. QED.

In the case where u and v are not adjacient in the graph, the union of the closures is: U := {xv, xu} union {yvw, yuw : w is in V\{u, v}, if such y's exist in X}. Then a clopen set in this is: {xv} union {{yvw : w is in V\{u, v}, if such y's exist in X}, since {xu} union {{yuw : w is in V\{u, v}, if such y's exist in X} is also open. So U is not connected.

### 2) **Proposition:** after a turn in a finite game, any coloured element of X is in a stone.

*Proof:* by induction on the number of turns played.

__Base:__ at the start of the game, by Rule 3, there are
no coloured elements of X, so they are all inside stones.

__Inductive step:__ Defn 9 tells us how the board is changed in a turn (either definition of turn).
When we have just coloured all elements of e one's own colour, all coloured elements outside of e are still in stones by induction.
All elements inside e are inside e, which is a stone.

When we have just cleared a colour C, we have coloured empty all elements in {x in X such that x is of colour C, and for all stones s of colour C in s : x is in s, string(s) does not have liberty}. If x in X is still coloured after this, then x was not in that set. I.e. x is not of colour C so it's in a stone by induction, or x is of colour C and there exists a stone s : x is in s, and string(s) has a liberty. I.e. x is in a stone. []

I'm not sure this result extends to games with countably infinite numbers of moves. I'll try to think up a counterexample.

### 3) **Proposition:** Given a stone s of colour C, string(s) is go-connected.

*Proof:* Let sets of type A be defined as in Defn 7 as the members of:

For all *A*, is T-connected since *A* is go-connected.

, since:

s is non-empty because the empty set is not in S, by Defn 1

So by a well known result of topology,

is T-connected, so by another well known result, is T-connected as well.

Consider:

If Q is T-connected, then by Defn 5, string(s) is go-connected. We will show that , so that Q is connected.

Show that

Show that

Let:

So:

But Q is closed, so

So Q is T-connected, so string(s) is go-connected.