I was wondering under what conditions will the quotient of I at its endpoints $\partial I$ with a simply connected space will give $\pi_1 = \mathbb{Z}$. To be clear, the end points do not need to quotient to the same point. I think I have a proof for $T_1$ spaces, and I'm having trouble of thinking up counter examples for non-$T_1$. Is there a standard proof for all spaces?
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If I understand your question correctly, I think a clearer phrasing of the operation you're asking about would be "gluing the endpoints of an interval to a simply connected space", or to be categorical about it, taking the pushout of the inclusion $\partial I\hookrightarrow I$ with a map $\partial I\to X$ where $X$ is simply connected – Zev Chonoles Jul 09 '13 at 20:02