Question

Let B in R^n be any set. Define:\(C=\{x\in R^n| d(x, y)<1 for some y \in B\}\).
Show that C is open
Please, help

Answer 1

Let H be your set. If B is empty then your set will be empty. If not
let a an element in H, then there is b in B so that d(a,b) < 1.
Let
\[
\epsilon = 1- d(a,b)
\]
Show that all elements that are in an open sphere of center a and radius\( \epsilon\)
are in H. Can you do that?

Answer 2

then let x such that \(d (x, a) < \epsilon\) then
\[d (x, b)\le d (x, a) + d (a, b) < \epsilon + 1 - \epsilon = 1\]
Hence x is in H.This will do it why?

Answer 3

H+C in my post above

Answer 4

Thank you very much. I got it.