let S be RealNormSpace; :: thesis: for X being Subset of S st ( for r being Point of S st r in X holds
ex N being Neighbourhood of r st N c= X ) holds
X is open
let X be Subset of S; :: thesis: ( ( for r being Point of S st r in X holds
ex N being Neighbourhood of r st N c= X ) implies X is open )
assume that
A1: for r being Point of S st r in X holds
ex N being Neighbourhood of r st N c= X and
A2: not X is open ; :: thesis: contradiction
not X ` is closed by A2;
then consider s1 being sequence of S such that
A3: rng s1 c= X ` and
A4: s1 is convergent and
A5: not lim s1 in X ` ;
consider N being Neighbourhood of lim s1 such that
A6: N c= X by A1, A5, SUBSET_1:29;
consider g being Real such that
A7: 0 < g and
A8: { y where y is Point of S : ||.(y - (lim s1)).|| < g } c= N by NFCONT_1:def 1;
consider n being Nat such that
A9: for m being Nat st n <= m holds
||.((s1 . m) - (lim s1)).|| < g by A4, A7, NORMSP_1:def 7;
n in NAT by ORDINAL1:def 12;
then n in dom s1 by FUNCT_2:def 1;
then A10: s1 . n in rng s1 by FUNCT_1:def 3;
||.((s1 . n) - (lim s1)).|| < g by A9;
then s1 . n in { y where y is Point of S : ||.(y - (lim s1)).|| < g } ;
then s1 . n in N by A8;
hence contradiction by A3, A6, A10, XBOOLE_0:def 5; :: thesis: verum