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, NFCONT_1:def 6;
then consider s1 being sequence of S such that
A3: ( rng s1 c= X ` & s1 is convergent & not lim s1 in X ` ) by NFCONT_1:def 5;
lim s1 in X by A3, SUBSET_1:50;
then consider N being Neighbourhood of lim s1 such that
A4: N c= X by A1;
consider g being Real such that
A5: 0 < g and
A6: { y where y is Point of S : ||.(y - (lim s1)).|| < g } c= N by NFCONT_1:def 3;
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, NORMSP_1:def 11;
||.((s1 . n) - (lim s1)).|| < g by A7;
then s1 . n in { y where y is Point of S : ||.(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