Topological Go is a variant allowing for many types of boards, including normal boards and Go on a Board Without Lines. It is described (though using some topological terminology) at https://web.archive.org/web/20080301141852/http://www.nrinstruments.demon.co.uk/topologo/topologo.html .
The board is described as a go-space, which consists of three sets: X contains every location on the board, T contains every open set of locations on the board, and S contains every possible stone. Elements of S must be open sets, and the empty set cannot be in S.
Being open is a topological term which is only needed here to define what a stone can be and when stones are connected. The only constraints on the notion of openness is that the empty set is open, the set of all points is open, that the union of any number of open sets is open, and that the intersection of any finite number of open sets is open. These constraints are of no importance when studying an existing go-space. Intuitively, an open set is a set whose border is not part of the set.
Every point in X is colored either black, white or empty. An element of S is said to be of a color if and only if all elements of it (remember, elements of S are themselves sets) are of that color. A stone is an element of S that is colored black or white, while a legal placement is an element of S that is colored empty.
The hardest definition to understand: formally, a set of stones is go-connected if and only if the closure of the union of the stones is connected.
Formally, the closure of a set s is the set of all points p such that every open set containing p contains a point of s. Intuitively, a closure is an open set plus its border. Formally, a set s is connected if there are no two non-overlapping open sets that s is contained in. Intuitively, a set is connected if it cannot be divided in two by a line that doesn't go through it.
Therefore, intuitively, two stones are go-connected if and only if their borders touch each other, and a set of more than two stones is go-connected if and only if all its stones are go-connected, directly or indirectly.
A go-connected set g of stones has liberty if and only if there exists a legal placement e such that adding e to g results in a go-connected set.
The string of a stone s of color c is the union of all sets of stones colored c that are go-connected and contain s.
Clearing a color consists of coloring empty all points x of that color such that x is in a string that does not have liberty.
A board play consists of choosing a legal placement, coloring it one's own color, clearing one's opponent's color, and clearing one's own color.
These rules only describe how to place and capture stones; scoring and ko must be handled by special rules, and what little remains can be described by normal rules.
On a normal board, X consists not only of points on the board but the connections between them. A set is open (that is, it's in T) if and only if if it contains a connection, then it also contains the stones it connects. S consists of all sets containing exactly one point on the board.
The intuitive notion of a border here corresponds to the connections around a set of points. Since an open set containing a connection must also contain the points next to it, taking the closure of a set adds all connections adjacent to it. (Note that closures usually are not open.)
The result is that a set s that is connected under the normal rules of Go is also connected here, because if a connection is given to one set, both its points must also be in that set, so s must go entirely to one set, making it go-connected. (Remember that the two open sets a go-connected set cannot be divided into can contain points that are not in that go-connected set.)
Here, X consists of all points on the board, T consists of all sets that are open under the usual topology over X, and S consists of all possible stones (the insides of circles excluding their borders, unless you like stones that aren't round). The results can be understood well enough using the intuitive definitions of everything; if you prefer being formal, T is constructed by starting with S, adding X and the empty set, and adding everything required by the rules of openness stated above.