In Willard's General Topology, he wrote that

By identiyfiny $(0,x)$ with $(2\pi,x)$... the corresponding quotient map of $[0,2 \pi] \times [0, 2\pi ] $ which gives the cylinder $S^1 \times [0,2\pi] $ as a quotient space is $f(x,y) = ((\cos x, \sin x), y)$.

We know this resulting quotient space is homeomorphic to the identification space of of our square. But the cylinder has the quoteint topology and not the usual topology it inherits as a subspace of $\mathbb{R}^3.$ Are these two topologies the same? If not, then what was the point of all this?

  • $\begingroup$ Is $f$ a homeomorphism ? $\endgroup$ – James S. Cook Feb 18 '17 at 19:07

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