Quotient space of the torus with two points identified - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T01:21:17Z http://mathoverflow.net/feeds/question/63848 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63848/quotient-space-of-the-torus-with-two-points-identified Quotient space of the torus with two points identified Beatty 2011-05-03T20:34:24Z 2011-05-03T20:34:24Z <p>Take the torus $T=S^{1} \times S^{1}$. Choose two points $x, y \in T$ and define a quotient topology by identifying $x$ and $y$. Let $X$ denote the quotient space. Prove that: a) Compute the fundamental group of $X$. b) Prove that $X$ is not homeomorphic to a surface.</p>