consider x being set such that
A7: x in the carrier of M by XBOOLE_0:def 1;
reconsider x = x as Point of M by A7;
set B = cl_Ball x,1;
take S = NAT --> (cl_Ball x,1); :: thesis: ( S is non-empty & S is bounded & S is closed )

A8: now

let y be set ; :: thesis: ( y in dom S implies not S . y is empty )
assume y in dom S ; :: thesis: not S . y is empty
then reconsider n = y as Element of NAT ;
A9: cl_Ball x,1 = S . n by FUNCOP_1:13;
dist x,x = 0 by METRIC_1:1;
hence not S . y is empty by A9, METRIC_1:13; :: thesis: verum

end;

A10: now

let i be natural number ; :: thesis: S . i is closed
i in NAT by ORDINAL1:def 13;
then A11: S . i = cl_Ball x,1 by FUNCOP_1:13;
([#] M) \ (cl_Ball x,1) in Family_open_set M by NAGATA_1:14;
then (cl_Ball x,1) ` is open by Def3;
hence S . i is closed by A11, Def4; :: thesis: verum

end;

now

let i be natural number ; :: thesis: S . i is bounded
A12: i in NAT by ORDINAL1:def 13;
cl_Ball x,1 is bounded by TOPREAL6:67;
hence S . i is bounded by A12, FUNCOP_1:13; :: thesis: verum

end;

hence ( S is non-empty & S is bounded & S is closed ) by A8, A10, Def1, Def8, FUNCT_1:def 15; :: thesis: verum