theorem Th7: :: TOPDIM_2:7

for I being Integer
for TM being metrizable TopSpace st TM is second-countable & TM is finite-ind & ind TM <= I holds
ex F being finite Subset-Family of TM st
( F is Cover of TM & F is finite-ind & ind F <= 0 & card F <= I + 1 & ( for A, B being Subset of TM st A in F & B in F & A meets B holds
A = B ) )