Let $I=[0,1]$ be the unit interval, viewed as a topological subspace of the real line $R$. Let $I_o=\left\{0,1 \right\}$ be the boundary of $I$. Then denote by $I/{I_o}$ the topological space defined by taking $I$ and shrinking $I_o$ to a point $a^*$ and the topology being the identification topology.
I read the statement "a basis for the open sets of $I/I_o$ containing $a^*$ is the totality of images of the sets of the form $[0,\epsilon) \cup (1-\epsilon,1]$".
How can we see that? Thanks :-)