let T be non empty TopSpace; :: thesis: for A being Subset of T
for F being Subset-Family of T st F = {A} holds
( A is closed iff F is closed )
let A be Subset of T; :: thesis: for F being Subset-Family of T st F = {A} holds
( A is closed iff F is closed )
let F be Subset-Family of T; :: thesis: ( F = {A} implies ( A is closed iff F is closed ) )
assume A1: F = {A} ; :: thesis: ( A is closed iff F is closed )

hereby :: thesis: ( F is closed implies A is closed )

assume A is closed ; :: thesis: F is closed
then for A being Subset of T st A in F holds
A is closed by A1, TARSKI:def 1;
hence F is closed by TOPS_2:def 2; :: thesis: verum

end;

assume A2: F is closed ; :: thesis: A is closed
A in F by A1, TARSKI:def 1;
hence A is closed by A2, TOPS_2:def 2; :: thesis: verum