Math Is Fun Forum

almost there

Member

Registered: 2009-11-11

Posts: 21

connected components in uniform topology

has the uniform topology (induced by the uniform metric). Show that x and y are in the same connected component if and only if x-y is bounded (as a sequence of real numbers, of course).

I can show that

is not connected by producing a separation into bounded and unbounded elements.

There is a hint in the book (Munkres), but again I do not see how it helps. Munkres says "it suffices to consider the case y=0". If I consider this case, then I see that if x is in the zero component, then x-0=x must be bounded (because [0], the zero component, is bounded and I have shown the separation above). However, I am unclear about the other direction. If x is bounded, why does this necessarily imply that x must be in the 0 component, [0]? And I still don't see how this special case is sufficient to prove the general result originally stated.

Thanks in advance.