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Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y is continuous if and only if nx º g and ty o g are both continuous.

Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y is continuous if and only if nx º g and ty o g are both continuous.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do I show 7.36? Could you explain this in great detail? Thank you!

Definition. Suppose T and T' are two topologies on the same underlying set X. If T 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 : X homeomorphism from X to f(X), where f(X) has the subspace topology from Y. Y is an embedding if and only if f : X f(X) is a Definition. The projection maps Ax : X×Y → by 7x(x, y) = x and ty(x, y) = y. X and Ay : X × Y → Y are defined Theorem 7.32. Let X and Y be topological spaces. The projection maps īx,Ty on X×Y are continuous, surjective, and open. In fact, the topology on the product space can be characterized as the coarsest topology that makes the projection maps continuous. Theorem 7.33. Let X and Y be topological spaces. The product topology on X x Y is the coarsest 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 is continuous if and only if nx og and ty o g are both continuous.
Transcribed Image Text:Definition. Suppose T and T' are two topologies on the same underlying set X. If T 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 : X homeomorphism from X to f(X), where f(X) has the subspace topology from Y. Y is an embedding if and only if f : X f(X) is a Definition. The projection maps Ax : X×Y → by 7x(x, y) = x and ty(x, y) = y. X and Ay : X × Y → Y are defined Theorem 7.32. Let X and Y be topological spaces. The projection maps īx,Ty on X×Y are continuous, surjective, and open. In fact, the topology on the product space can be characterized as the coarsest topology that makes the projection maps continuous. Theorem 7.33. Let X and Y be topological spaces. The product topology on X x Y is the coarsest 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 is continuous if and only if nx og and ty o g are both continuous.
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