% problem-set/topology/point.set/lemma12.clauses
% created : 08/08/88
% revised : 08/17/89

% description : 
% 
% lemma 12: Let (x,xt) be a topological space.  Then 0 and x are 
%	closed in x; for y1, y2 closed, y1 intersect y2 is closed 
%	and y1 union y2 is closed.

% representation : 
%
% See axioms.decl for a full explanation.

% denial: 

top_space(X,T).
-closed(0,X,T) | -closed(X,X,T) | closed(Y1,X,T).
-closed(0,X,T) | -closed(X,X,T) | closed(Y2,X,T).
-closed(0,X,T) | -closed(X,X,T) | -closed(inter_s(Y1,Y2),X,T) | 
	-closed(join_s(Y1,Y2),X,T).