let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X1,X2 constitute_a_decomposition holds

( X1 is open iff X2 is closed )

let X1, X2 be non empty SubSpace of X; :: thesis: ( X1,X2 constitute_a_decomposition implies ( X1 is open iff X2 is closed ) )

reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;

assume A1: for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds

A1,A2 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( X1 is open iff X2 is closed )

thus ( X1 is open implies X2 is closed ) :: thesis: ( X2 is closed implies X1 is open )

A2 is closed ; :: according to BORSUK_1:def 11 :: thesis: X1 is open

( X1 is open iff X2 is closed )

let X1, X2 be non empty SubSpace of X; :: thesis: ( X1,X2 constitute_a_decomposition implies ( X1 is open iff X2 is closed ) )

reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;

assume A1: for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds

A1,A2 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( X1 is open iff X2 is closed )

thus ( X1 is open implies X2 is closed ) :: thesis: ( X2 is closed implies X1 is open )

proof

assume A4:
for A2 being Subset of X st A2 = the carrier of X2 holds
assume A2:
for A1 being Subset of X st A1 = the carrier of X1 holds

A1 is open ; :: according to TSEP_1:def 1 :: thesis: X2 is closed

end;A1 is open ; :: according to TSEP_1:def 1 :: thesis: X2 is closed

now :: thesis: for A2 being Subset of X st A2 = the carrier of X2 holds

A2 is closed

hence
X2 is closed
; :: thesis: verumA2 is closed

let A2 be Subset of X; :: thesis: ( A2 = the carrier of X2 implies A2 is closed )

assume A2 = the carrier of X2 ; :: thesis: A2 is closed

then A3: A1,A2 constitute_a_decomposition by A1;

A1 is open by A2;

hence A2 is closed by A3, Th11; :: thesis: verum

end;assume A2 = the carrier of X2 ; :: thesis: A2 is closed

then A3: A1,A2 constitute_a_decomposition by A1;

A1 is open by A2;

hence A2 is closed by A3, Th11; :: thesis: verum

A2 is closed ; :: according to BORSUK_1:def 11 :: thesis: X1 is open

now :: thesis: for A1 being Subset of X st A1 = the carrier of X1 holds

A1 is open

hence
X1 is open
; :: thesis: verumA1 is open

let A1 be Subset of X; :: thesis: ( A1 = the carrier of X1 implies A1 is open )

assume A1 = the carrier of X1 ; :: thesis: A1 is open

then A5: A1,A2 constitute_a_decomposition by A1;

A2 is closed by A4;

hence A1 is open by A5, Th11; :: thesis: verum

end;assume A1 = the carrier of X1 ; :: thesis: A1 is open

then A5: A1,A2 constitute_a_decomposition by A1;

A2 is closed by A4;

hence A1 is open by A5, Th11; :: thesis: verum