consider x being set such that
A1: x in the carrier of M by XBOOLE_0:def 1;
reconsider x = x as Point of M by A1;
set B = Ball x,1;
take S = NAT --> (Ball x,1); :: thesis: ( S is non-empty & S is bounded & S is open )

A3: now

let i be natural number ; :: thesis: S . i is bounded
( Ball x,1 is bounded & i in NAT ) by ORDINAL1:def 13, TBSP_1:19;
hence S . i is bounded by FUNCOP_1:13; :: thesis: verum

end;

A4: now

let y be set ; :: thesis: ( y in dom S implies not S . y is empty )
assume A5: y in dom S ; :: thesis: not S . y is empty
reconsider n = y as Element of NAT by A5;
dist x,x = 0 by METRIC_1:1;
then ( x in Ball x,1 & Ball x,1 = S . n ) by FUNCOP_1:13, METRIC_1:12;
hence not S . y is empty ; :: thesis: verum

end;

now

let i be natural number ; :: thesis: S . i is open
i in NAT by ORDINAL1:def 13;
then ( S . i = Ball x,1 & Ball x,1 in Family_open_set M ) by FUNCOP_1:13, PCOMPS_1:33;
hence S . i is open by Def3; :: thesis: verum

end;

hence ( S is non-empty & S is bounded & S is open ) by A3, A4, Def1, Def7, FUNCT_1:def 15; :: thesis: verum