How do you evaluate the continuity of a function with product topology general topology continuity metric spaces math?
In general topology, we often encounter functions defined on product spaces. A product space is a Cartesian product of multiple topological spaces, and the product topology on the product space is the finest topology that makes all the projection maps continuous.
To evaluate the continuity of a function on a product space with product topology, we can use the following theorem:
Theorem: A function f: X x Y -> Z is continuous if and only if f is continuous in each variable separately.
In other words, to show that f is continuous, we need to show that for any open set U in Z, the preimage f^-1(U) is open in X x Y. This can be done by showing that the preimage of U under each projection map is open in the corresponding space.
Example:
Consider the function f: R x R -> R defined by f(x, y) = xy. We want to show that f is continuous.
Let U be an open set in R. Then the preimage of U under the projection map p1: R x R -> R is p1^-1(U) = { (x, y) | xy in U }. This is open in R x R because it is the inverse image of an open set under a continuous map.
Similarly, the preimage of U under the projection map p2: R x R -> R is p2^-1(U) = { (x, y) | xy in U }. This is also open in R x R.
Therefore, by the theorem above, f is continuous.
Related Questions:
- What is the product topology?
- The product topology on a product space is the finest topology that makes all the projection maps continuous.
- What is the theorem for evaluating the continuity of a function on a product space?
- A function f: X x Y -> Z is continuous if and only if f is continuous in each variable separately.
- What is an example of a continuous function on a product space?
- The function f(x, y) = xy is continuous on R x R.
- What is the preimage of a set under a function?
- The preimage of a set U under a function f is the set { x | f(x) in U }.
- What is an open set in a topological space?
- An open set in a topological space is a set that contains an open neighborhood of each of its points.
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