Given \(I=[0,1]\) and the equivalence relation on \(I\) generated by \(0\sim 1\), i.e. \(x\sim y\) if and only if \(x=y\) or \(\{x,y\}=\{0,1\}\).
Then the quotient space \(I/\sim\) is homeomorphic to the unit circle \(\mathbb{S}^1\).
Proof
First, consider the map \(\omega\colon I\to \mathbb{S}^1\) given by \(\omega(s)=\e^{2\pi \i s}\). This is a quotient map (see [1, Example 3.66]), which can be checked by definition.
Then \(\omega\) makes the same identifications as \(\sim\) and \(\mathbb{S}^1\) is homeomorphic to \(I/\sim\) by (0x67b024f3) .
References Link to heading
- J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7