Question

TOPOLOGY

We have in $\mathbb{R}$ the topology generated by the basis $B_{\leftarrow }=\left \{ \left ( -\infty ,b \right ):b\in \mathbb{R}\left. \right \}\right.$ , which is not the usual topology.

Find the closure and the interior of the set A=(0,1) in this topology.

Answer 1

Let R is with the topology generated by basis B = { (-infinity, b): b element of 1R} then interior of (0, 1) = phi as no open set (-infinity, a) is contained in A = (0, 1) for all a element of 1R closure of (0, 1) = [0, infinity) as for each a element of [0, infinity) every open set containing a intersect A