How do I show 7.33? Could you explain this in great detail? Thank you! Definition. Suppose T and T' are two topologies on the same underlying set X. IfT C T', then we say T' is finer than T. Alternatively, we say T is coarser than J'.We say strictly coarser or strictly finer if additionally J # T".Definition. A function f : Xhomeomorphism from X to f(X), where f(X) has the subspace topology from Y.Y is an embedding if and only if f : Xf(X) is aDefinition. The projection maps Ax : X×Y →by 7x(x, y) = x and ty(x, y) = y.X and Ay : X × Y → Y are definedTheorem 7.32. Let X and Y be topological spaces. The projection maps īx,Ty on X×Yare continuous, surjective, and open.In fact, the topology on the product space can be characterized as the coarsesttopology that makes the projection maps continuous.Theorem 7.33. Let X and Y be topological spaces. The product topology on X x Y is thecoarsest topology on X × Y that makes the projection maps T,Ty on X × Y continuous.Theorem 7.35. Let X and Y be topological spaces. For every y E Y, the subspace X x{y}of X × Y is homeomorphic to X.Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y iscontinuous if and only if nx og and ty o g are both continuous.

Name: How do I show 7.33? Could you explain this in great detail? Thank you!
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: How do I show 7.33? Could you explain this in great detail? Thank you! Definition. Suppose T and T' are two topologies on the same underlying set X. IfT C T', then we say T' is finer than T. Alternatively, we say T is coarser than J'.We say strictly