Some questions about topology in in $[0,1)$ and $\Pi = \{z \in \mathbb{C}: |z|=1\}$

Question

I have to answer some questions and show some stuff concerning the topological properties of $[0,1)$ and $\Pi = \{z \in \mathbb{C}: |z|=1\}$. As I am a newbie in topology, I have some problems with it.

First I have to name some topological properties of the quotient space $(\mathbb{R\backslash Z},\theta)$ where $\theta$ denotes the induced topology from the projection $ x \mapsto x + \mathbb{Z}$. I already know that it is compact and Hausdorff. Are there any other important properties? And can I describe $\tau$ any "better" than its elementes are exactly those $U$ for which $\cup_{z\in\mathbb{Z}}\{z+x:x\in U\}$ is open in $\mathbb{R}$?

Second, is there a countable basis for $\Pi$ equipped with the subspace topology? I was thinking that the topology $\mathbb{R}$ has a countable basis with all the open intervalls with border points in $\mathbb{Q}$, maybe $\{ (q_{i_1},q_{j_1})\times(q_{i_2},q_{j_2})\cap \Pi:q_{k_l}\in\mathbb{Q},l\in\{1,2\},k\in\mathbb{N}\}$ is a basis for $\Pi$? (Considering $\mathbb{C}$ as a vector space).

Last, I am asked to find a topology on $[0,1)$ with the following properties: $$(i): \phi:\Pi \rightarrow [0,1), \phi(e^{2\pi ix})=x, 0\leq x < 1, \text{is $\tau_{\Pi}-\tau$-continuous}$$ $$(ii) \text{ the topological space $([0,1),\tau)$ is compact}$$ and which properties does this topological space additionally have?

Since $\phi$ is surjective I was thinking of defining $\tau$ as the set $\{\phi^{-1}(U):U \text{ is open in }\tau_{\Pi}\}$. Then it would clearly be continous. However, neither am I sure whether it is compact, nor do I know how to prove it.

Thanks for your help.

Answer 1

1. $\mathbf{R}/\mathbf{Z}$ is metrizable with the metric \begin{align}d(x + \mathbf{Z}, y + \mathbf{Z}) &= \min\{|x' - y'| : x' \in x + \mathbf{Z}, y' \in y + \mathbf{Z}\} \\ &=\min\{|x-y+n| : n \in \mathbf{Z}\}. \end{align} This means that it has every topological property that a compact metric space has. Additionally, every point of $\mathbf{R}/\mathbf{Z}$ has a neighbourhood which is homeomorphic to $\mathbf{R}$.

2. Yes, every subspace of a second countable space is also second countable by the same argument you give.

3. The topology you have in mind is $\tau = \{ U \subseteq [0,1) : \phi^{-1}(U) \in \tau_{\Pi} \}$. What you've written ($U$ is open in $\Pi$) doesn't make sense because $U \subseteq [0,1)$ not $\Pi$. Show that with your topology, $\phi : \Pi \to [0, 1)$ is a homeomorphism. Then $[0,1)$ is compact because $\Pi$ is compact.