let X be Subset of COMPLEX ; :: thesis: ( ( for z0 being Complex st z0 in X holds
ex N being Neighbourhood of z0 st N c= X ) implies X is open )
assume that
A1: for z0 being Complex st z0 in X holds
ex N being Neighbourhood of z0 st N c= X and
A2: not X is open ; :: thesis: contradiction
not X ` is closed by A2, Def11;
then consider s1 being Function of NAT ,COMPLEX such that
A3: ( rng s1 c= X ` & s1 is convergent & not lim s1 in X ` ) by Def10;
consider N being Neighbourhood of lim s1 such that
A4: N c= X by A1, A3, SUBSET_1:50;
consider g being Real such that
A5: 0 < g and
A6: { y where y is Complex : |.(y - (lim s1)).| < g } c= N by Def5;
consider n being Element of NAT such that
A7: for m being Element of NAT st n <= m holds
|.((s1 . m) - (lim s1)).| < g by A3, A5, COMSEQ_2:def 5;
|.((s1 . n) - (lim s1)).| < g by A7;
then s1 . n in { y where y is Complex : |.(y - (lim s1)).| < g } ;
then A8: s1 . n in N by A6;
n in NAT ;
then n in dom s1 by FUNCT_2:def 1;
then s1 . n in rng s1 by FUNCT_1:def 5;
hence contradiction by A3, A4, A8, XBOOLE_0:def 5; :: thesis: verum