How to show any space is dually scattered? As the title explains, how to show any space is dually scattered? > A topological space $X$ is _dually scattered_ if for any neighbourhood assignment $\\{ O_x : x \in X \\}$ there is a scattered subspace $Y \subseteq X$ such that $\bigcup_{x \in Y} O_x = X$.) Thanks ahead:)

Let $\\{ O (x) : x \in X \\}$ be any neighbourhood assignment of the space $X$. For as long as possible inductively construct a (transfinite) sequence $\\{ x_\alpha : \alpha < \eta \\}$ in $X$ so that

* $x_\alpha \
otin \bigcup_{\xi < \alpha} O (x_\xi)$.



(If $\\{ O ( x_\xi ) : \xi < \alpha \\}$ does not cover $X$ take any $x_\alpha \in X \setminus \bigcup_{\xi < \alpha} O ( x_\xi )$.)

Letting $Y = \\{ x_\alpha : \alpha < \eta \\}$, note that by construction we will have that $\bigcup_{\alpha < \eta} O ( x_\alpha ) = X$, so it suffices to show that $Y$ is scattered. But note that if $A \subseteq \eta$ is nonempty and $\alpha_0 = \min (A)$ then $x_{\alpha_0}$, is an isolated point of $\\{ x_\alpha : \alpha \in A \\}$. (By construction we have that $O( x_{\alpha}) \cap Y \subseteq \\{ x_\xi : \xi \leq \alpha \\}$.)