Suppose that f_k -> f uniformly on (0,1). Let 0 < x < 1. Suppose that lim f_k(t) = A_k for k=1,2,... Show that {A_k} converges and lim f(t) = LIM A_k. That is show lim LIM f_k(t) = LIM lim f_k(t). Where lim represents the limit as t approaches x and LIM represents the limit as k approaches infinity.
Real valued fucntion on a topological space
Hi, I have a question about showing the continuity of real valued function on a topological space. Could anyone help me with detail explanation? Question as attached. Thanks a lot.
a) Construct a scatter diagram of the number of crimes and police expenditures per capita, with number of crimes as the predictor variable. What can you say about the relationship between these two variables based on the scatter plot? b) Find the least-squares regression equation that predicts police expenditures per capita fro ...continues
Show that f is proper if and only if f* is continuous
Let X and Y be locally compact Hausdorff spaces. Let X* and Y* be their one point compactifications. Let f be a continuous map from X to Y. Let f* be the map from X* to Y* whose restriction to X is f, and which takes the point at infinity in X* to the point at infinity in Y*. Show that f is proper if and only if f* is contin ...continues
Prove that the empty set is a subset of every set.
Prove that the empty set is a subset of every set.
For x an element of R upper k and y an element of R upper k define
For x an element of R upper k and y an element of R upper k define d1 (x, y) = max{|x of i – y of i| : 1≤i≤k} and d2 (x, y) = min{|x of i – y of i| : 1≤i≤k}. Determine for each of these whether it is a metric or not. Fully explain your answers. Thank you so much.
For any finite collection G1, G2, ...., Gn of open sets, intersection
Give examples to show that the finiteness of the collections in parts c and d is essential. c) for any finite collection G1, G2, ...., Gn of open sets, intersection (at the top of the intersection sign is n and at the bottom is i=1) of Gi is open. d) For any finite collection F1, F2, ...., Fn of closed sets, union sign ( ...continues
Does the closure of a union equal the union of the closures?
See attached file.
Let A and B be separated subsets of some Rk, suppose
Please see the attach file. Let A and B be separated subsets of some Rk, suppose and , and define for . Put , . [Thus if and only if .] (a) Prove that Ao and Bo are separated subsets of R1 . (b) Prove that there exists such that . (c) Prove that every convex subset of Rk is conne ...continues
Give an example in R2 of a convex set.
Give an example in R2 of a convex set. ( I know that an example of a convex set is something like a circle or something just as long as you can draw a segment between and 2 pts and the segment still be in the circle, I think, I know something like the packman shape or a peanut is NOT convex) Anyway my teacher wants the answe ...continues
