Exercise 3.1. Here is the unseen exercise done in the week 03 tutorial.

Consider the following topological spaces (the first five are viewed as subspaces of the Euclidean space). Determine which of these spaces are homeomorphic. Give a convincing description, or an explicit formula, for the homeomorphism where necessary; give reasons when the spaces are not homeomorphic.

1. The punctured plane \(\RR ^2\setminus \{(0,0)\}\)

2. The open annulus \(A = \{(x,y)\in \RR ^2: 1<\sqrt {x^2+y^2}<2\}\)

3. The twice-punctured sphere \({S^2\setminus \{N,S\}},\) where \({S^2= \{(x,y,z)\in \RR ^3: x^2+y^2+z^2=1\}},\) \(N=(0,0,1)\) and \(S=(0,0,-1)\)

4. The punctured open hemisphere \((S^2 \cap \{z<0\})\setminus \{S\}\)

5. The (infinite) cylinder \(C=\{(x,y,z)\in \RR ^3: x^2+y^2=1\}\)

6. and finally, the set \(\RR ^2\) with antidiscrete topology.