let PM be MetrStruct ; :: thesis: TopStruct(# the carrier of PM,(Family_open_set PM) #) is TopSpace
set T = TopStruct(# the carrier of PM,(Family_open_set PM) #);
A1: the carrier of TopStruct(# the carrier of PM,(Family_open_set PM) #) in the topology of TopStruct(# the carrier of PM,(Family_open_set PM) #) by Th34;
A2: for a being Subset-Family of TopStruct(# the carrier of PM,(Family_open_set PM) #) st a c= the topology of TopStruct(# the carrier of PM,(Family_open_set PM) #) holds
union a in the topology of TopStruct(# the carrier of PM,(Family_open_set PM) #) by Th36;
for p, q being Subset of TopStruct(# the carrier of PM,(Family_open_set PM) #) st p in the topology of TopStruct(# the carrier of PM,(Family_open_set PM) #) & q in the topology of TopStruct(# the carrier of PM,(Family_open_set PM) #) holds
p /\ q in the topology of TopStruct(# the carrier of PM,(Family_open_set PM) #) by Th35;
hence TopStruct(# the carrier of PM,(Family_open_set PM) #) is TopSpace by A1, A2, PRE_TOPC:def 1; :: thesis: verum