Topology Closed Neighborhood Proof

(8) N is a closed neighborhood of p in R_U iff N=R

Where R represents the real numbers and

R_U is the upper topological space for the real numbers (R, T_U)

T_U = {A is a subset of R | a is an element of A implies (a - e, +inf) is a subset of A for some e > 0} (upper topology)

I proved "=>"

Need help showing "<="

ie: If N=R, then N is a closed neighborhood of p in R_U

Re: Topology Closed Neighborhood Proof

The topology of is the set .

The direction you did was the "hard" direction. I think you forgot that the whole space is always a closed set, since it's the complement of the empty set, which is always open.

Let . Then , which is open. Let . Will show is a closed neighborhood of .

Have that is closed. Since , it follows that is a closed neighborhood of .