I dont really understand about quotient space. I read this paper https://eudml.org/doc/214180. In that paper, Fleischman defined a space $Y$ as follows: Let $W(a)$ be the set of all ordinals which precede $a$ and $X$ be the topological sum of countably many copies of the space $W(\omega_1 +1)$. Define space $Y$ as the quotient space of $X$ in which all equal to countable ordinals are identified.
My question is:
1.Why we can consider $Y$ as $\{a_n: n \in \mathbb{N}\} \cup W(\omega_1)$ with the basic neighbourhoods of a point of $W(\omega_1)$ are the open intervals of $W(\omega_1)$ to which it belongs, while the basic neighbourhoods of a point $a_n$ are the set $\{a_n\} \cup \{\beta \in W(\omega_1): \beta \geq \alpha\}$ for $\alpha \in W(\omega_1)$?
- Since no sequence is cofinal in $W(\omega_1)$, there is an $a \in W(\omega_1)$ with $a_n \leq a$ for all $n \in \mathbb{N}$. Why the subspace $\{\gamma \in W(\omega_1): \gamma \leq a\}$ of $Y$ is compact?
Is $\mathcal{U} = \{\alpha: \alpha < \omega_0\} \cup (\{a_1\} \cup [a, \omega_1)$ open cover of $\{\gamma \in W(\omega_1): \gamma \leq a\}$ that doesn't have finite subcover?