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 open iff F is open )

let A be Subset of T; :: thesis: for F being Subset-Family of T st F = {A} holds

( A is open iff F is open )

let F be Subset-Family of T; :: thesis: ( F = {A} implies ( A is open iff F is open ) )

assume A1: F = {A} ; :: thesis: ( A is open iff F is open )

A in F by A1, TARSKI:def 1;

hence A is open by A2, TOPS_2:def 1; :: thesis: verum

for F being Subset-Family of T st F = {A} holds

( A is open iff F is open )

let A be Subset of T; :: thesis: for F being Subset-Family of T st F = {A} holds

( A is open iff F is open )

let F be Subset-Family of T; :: thesis: ( F = {A} implies ( A is open iff F is open ) )

assume A1: F = {A} ; :: thesis: ( A is open iff F is open )

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

assume A2:
F is open
; :: thesis: A is open assume
A is open
; :: thesis: F is open

then for A being Subset of T st A in F holds

A is open by A1, TARSKI:def 1;

hence F is open by TOPS_2:def 1; :: thesis: verum

end;then for A being Subset of T st A in F holds

A is open by A1, TARSKI:def 1;

hence F is open by TOPS_2:def 1; :: thesis: verum

A in F by A1, TARSKI:def 1;

hence A is open by A2, TOPS_2:def 1; :: thesis: verum