I'm working through Hatcher's 3m notes. I'm reading a section where he is explaining some of the structure of horizontal essential surfaces. Let $M$ be a Seifert fibered 3-manifold and let $S$ be a horizontal 2-sided essential surface. Then, since $M$ is a disjoint union of circles, each of which meeting $S$ some number of times, we have that $M|S$ is a disjoint union of intervals $I$. So by crushing each $I$ to a point we see that there is a surface $T$ so that $M|S\to T$ has an $I-$bundle structure, with local triviality coming from the Seifert Fibering. Here's my question: why does $M|S$ being connected imply that
1) $T$ is connected
and
2) $M|S=S\times I$
While I believe the claim, the proof is escaping me. This seems like something elementary.