- Let X Be Any Set If T Is A Topology On X And T Is Another Topology Such That T T That Is Every Open Set With Res 1 (121.73 KiB) Viewed 33 times
Let X be any set. If T is a topology on X and T' is another topology, such that T ≤ T' (that is, every open set with respect to T is also open with respect to T', but the converse need not hold), then we say that T is coarser than T' and T' is finer than T. Now let T₁ and T₂ be two arbitrary topologies on X and show the following: Let T = TinT₂ (which you have already shown to be a topology in the the first book problem). Then 7 is the finest (that is, largest) topology that is coarser than both T₁ and T₂. II. Let T be defined as follows: U € T exactly if there are families of open sets, (U1,ª)µ€▲ in T₁ and (U2,α)µ€A in T2, such that U = U (U₁,a^U2,a). αEA Show that I is a topology and that T is finer than both T₁ and T₂. Then show that is the coarsest (that is, smallest) topology that is finer than both T₁ and T2.