[0,∞) is an unbounded closed subset of R, and (0,1) is a bounded open subset... so no.
Or if you want to think more geometrically, consider the plane R2. Then the x-axis is an unbounded closed set, and the open unit disc {x:∥x∥<1} is a bounded open set.
[0,∞) is an unbounded closed subset of R, and (0,1) is a bounded open subset... so no.
Or if you want to think more geometrically, consider the plane R2. Then the x-axis is an unbounded closed set, and the open unit disc {x:∥x∥<1} is a bounded open set.
After this post, it is clear I have embarrassed myself with this question Cheers for the answers.