In Topology, the second edition by Munkres, in section 22, on page 139 he says the following:
"Let $X$ be the closed unit ball
$$\{x\times y\mid x^2+y^2\leq 1\}$$
in $\mathbb{R}^2$, and let $X^*$ be the partition of $X$ consisting of all the one-point sets $\{x\times y\}$ for which $x^2+y^2<1$, along with the set $S^1=\{x\times y\mid x^2+y^2=1\}$. Typical saturated open sets in $X$ are pictured by the shaded regions in Figure 22.4. One can show that $X^*$ is homeomorphic with the subspace of $\mathbb{R}^3$ called the unit 2-sphere, defined by $$S^2=\left \{ \left ( x,y,z \right )\mid x^2+y^2+z^2=1 \right \}"$$
My question is: $U$ is a torus, why is $p(U)$ a disc? I really don't understand the idea of this picture. Can someone help me? Thanks.
Best Answer
The black dot in $p(U)$ is the class of the boundary that is squeezed to a point and the area $U$ is thus really a small 2-d area around that point, as drawn. The dot inside of $P(V)$ corresponds to the centre of the small disk $V$..