Are the following pairs of spaces homeomorphic?

(i) $mathbb Q cup (0, 1)$ and $mathbb R setminus mathbb Q$ (as subspaces of $(mathbb R, tau_{usual}))$;

(ii) $(mathbb C, tau_{disc})$ and $(mathbb C, tau_{cocountable})$;

(iii) a circle with one point removed (in $mathbb R^2$ with $tau_{usual}$) and $(mathbb R, tau_{usual})$;

(iv) $mathbb N$ with topology ${1, 2}, {3, 4}, {5, 6}, {7, 8}, ldots$ and all unions of these, and $mathbb N$ with topology ${1, 3}, {2, 4}, {5, 7}, {6, 8}, {9, 11}, {10, 12}, ldots$ and all unions of these.

The following have answers

(v) $(mathbb R, tau_{cocountable}), (mathbb R, tau_{cofinite})$

(vi) $mathbb R^2$ and the surface of a sphere with one point removed (natural metric

topologies here)

(vii) $(mathbb Q,l_x), (mathbb Q, l_y)$, where $x, y$ are two distinct rational numbers and $l_p$ is included-point topology

Book’s answers to (v) and (vii):

(v) No. The statement ‘there is a countably infinite closed subset’ is

(obviously) a homeomorphic invariant, is true in $(mathbb R, tau_{cocountable})$ and is false in $(mathbb R, tau_{cofinite})$

(vii) Yes. The map $h : mathbb Q → mathbb Q$ given by $h(x) = y, h(y) = x, h(z) = z$

when $z ne x$ or $y$ is routinely checkable to be a homeomorphism.

**My answers/questions regarding question above**:

(i) $mathbb Q, (0, 1)$ are not compact and so their union is not compact meaning there’s no continuous inverse function from $mathbb Q cup (0, 1)$

(ii) Any function from a discrete space to an indiscrete space is continuous. Since idenity $i$ is continuous and bijective, $i$ should work here to show homeomorphism

(iii) I think this is a special case of Stereographic projection

(iv) Any inverse image of an open set in $mathbb N$ is a union of $2$-sets. Since the given $2$-sets are open, their union is open as well. This works in both directions, so a continuous function with its continuous inverse likely exists. I am not sure how to find a concrete homeomorphism, but would $f(x, y) = (x, y – 1)$ if $y$ is odd and $f(x, y) = (x + 1, y)$ if $x$ is even work?

Are my answers to (i) through (iv) correct? If not (or if incomplete), how do I improve them?

(v) This question might be trivial, but I am still new to the very basics of topology. Consider $f: (mathbb R, tau_{cofinite}) to (mathbb R, tau_{cocountable})$. Every $Y in (mathbb R, tau_{cocountable})$ must have a pre-image $X in (mathbb R, tau_{cofinite})$. By definition, every $X$ is open, but for $f$ to be continuous $X$ must be cocountable. Correct?

(vi) It’s yet another special case of Stereographic projection. Correct?

(vii) Is $h(y)$ a typo? Did they mean $h^{-1}(y)?$

Thanks.