# Generating the Sorgenfrey topology by mappings into $\{0,1\}$, and on continuous images of the Sorgenfrey line

1. Show that the topology of the Sorgenfrey line can be generated be a family of mappings into a two-point discrete space.

2. Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$ but cannot be mapped onto $D(\mathfrak{c})$ (where $D(\kappa)$ is the discrete space of cardinality $\kappa$).

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1. For each real $a$ define a function $f_a : \mathbb{R} \to \{ 0,1 \}$ by $$f_a (x) = \begin{cases}0, &\text{if } x < a \\ 1, &\text{if }x \geq a.\end{cases}$$ Show that the topology on $\mathbb{R}$ generated by this family of functions is the Sorgenfrey line.
2. (a) Consider $\mathbb{Z}$ with the discrete topology. Define $f : \mathbb{R} \to \mathbb{Z}$ by $f (x) = \lfloor x \rfloor$ (where $\lfloor x \rfloor$, the floor of $x$, denotes the greatest integer not (strictly) greater than $x$). Show that this is continuous (and onto).
(b) Note that the Sorgenfrey line is separable (consider $\mathbb{Q}$, the set of rationals), and recall that any continuous image of a separable space is separable. Is $D(\mathfrak{c})$ separable?