Question

In the following questions, suppose the metric space is R with the standard metric.

For each set state if A is , open in B, closed in B, open and closed (clopen) in B or neither in subspace topology with the sets given.
a. If A = (0,1) and B = (0;[infinity]); then A is

Answer 1

In the subspace topology, A = (0,1) is open but not closed in B = (0;[infinity]).

Step-by-step explanation:

In the subspace topology, we need to consider the open sets of the metric space that intersect with the set A. For A = (0,1) and B = (0;[infinity]), the set A is open in B because for any point x in A, we can find an open interval around x that lies entirely within A. However, A is not closed in B because the complement of A, which is [1,∞), is not open in B.