Describe or draw a picture of the resulting quotient
space of:
1) The interval $[0, 4]$, as a subspace of $\mathbb R^1$, with integer points identified with
each other.
-I'm not sure on this, but since $[0,4]$ is a subspace topology I want to say that the end points $\{0\}$ and $\{4\}$ are closed, so the resulting space would be:
$\{1,2,3,4\}, \{2\}, \{3\}$, and $\{2,3\}$, right? Or would it look like something else?
and
2) The plane $\mathbb R^2$ with the circle $S^1$ collapsed to a point.
-For this I want to say it's just a single point or a sphere?
Any help would be great.
Best Answer
If you identify the integer points of $[0,4]$ and leave everything else alone, you simply collapse $0,1,2,3$, and $4$ to a single point, leaving everything else alone. The resulting quotient can be pictured like this:
The point at the bottom where the four loops meet is the collapsed $\{0,1,2,3,4\}$. The loops minus this point are the open intervals $(0,1),(1,2),(2,3)$, and $(3,4)$.
For the second problem, if you collapse $S^1$ to a point, you’re pinching off a bubble whose surface is that point together with the points of the open disk of radius $1$ centred at the origin. Picture a spherical surface lying on the plane: the point of tangency is the original $S^1$, the rest of the spherical surface is what used to be the open disk inside $S^1$, and the rest of the plane is the part of the original plane outside of $S^1$.
Perhaps it would help to imagine putting your fingertip at the origin under the plane. Now push up while shrinking the unit circle; when the unit circle has shrunk to a point — be sure to remove your finger first! — that point together with the part of the plane outside the original unit circle will now occupy the whole of the original plane, and that bubble that you started by pushing up with your fingertip will have been squeezed off into a spherical surface tangent to the plane at that point.