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 (with n at the top of the sign and i=1 at the bottom) of Fi is closed. I hope you can understand how I wrote the problems. I already did the proofs for the theorem that has to do with these sets. I'm just really having a hard time figuring out how to come up with a example.

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 connected

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 answer is the form {(x,y):??????} I don't know how to put it into that form. Also if you can I'm having the same problem for giving an example of a connected set in R2 in the same form.