let X be TopSpace; :: thesis: ( X is almost_discrete iff for A being Subset of X st A is open holds
Cl A = A )
thus ( X is almost_discrete implies for A being Subset of X st A is open holds
Cl A = A ) by Th21, PRE_TOPC:22; :: thesis: ( ( for A being Subset of X st A is open holds
Cl A = A ) implies X is almost_discrete )
assume A1: for A being Subset of X st A is open holds
Cl A = A ; :: thesis: X is almost_discrete

now :: thesis: for A being Subset of X st A is open holds
A is closed

let A be Subset of X; :: thesis: ( A is open implies A is closed )
assume A is open ; :: thesis: A is closed
then Cl A = A by A1;
hence A is closed ; :: thesis: verum

end;

hence X is almost_discrete by Th21; :: thesis: verum