let X be non empty TopSpace; :: thesis: for x being Point of X holds x in qComponent_of x

let x be Point of X; :: thesis: x in qComponent_of x

consider F being Subset-Family of X such that

A1: for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) and

A2: qComponent_of x = meet F by Def7;

( F <> {} & ( for A being set st A in F holds

x in A ) ) by A1, Th22;

hence x in qComponent_of x by A2, SETFAM_1:def 1; :: thesis: verum

let x be Point of X; :: thesis: x in qComponent_of x

consider F being Subset-Family of X such that

A1: for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) and

A2: qComponent_of x = meet F by Def7;

( F <> {} & ( for A being set st A in F holds

x in A ) ) by A1, Th22;

hence x in qComponent_of x by A2, SETFAM_1:def 1; :: thesis: verum