theorem Th22: :: TOPDIM_2:22

for TM being metrizable TopSpace
for A being Subset of TM st TM | A is second-countable & TM | A is finite-ind & ind A <= 0 holds
for F being finite Subset-Family of TM st F is open & F is Cover of A holds
ex g being Function of F,(bool the carrier of TM) st
( rng g is open & rng g is Cover of A & ( for a being set st a in F holds
g . a c= a ) & ( for a, b being set st a in F & b in F & a <> b holds
g . a misses g . b ) )