Two questions on Sorgenfrey line

Question

Show that the topology of the Sorgenfrey line can be generated be a family of mappings into a two-point descrete space.
Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$ but cannot be mapped onto $D(\mathfrak{c})$, where $D(\cdot)$ is the density of the topological space.

These appear to be questions from Engelking's text. If so, then for any cardinal number $\kappa$ by $D(\kappa)$ he denotes "the" discrete topological space whose underlying set has cardinality $\kappa$.

Arthur Fischer · Accepted Answer · 2012-06-24 07:02:25Z

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.
(The answers/hints to (2) will be with $D(\kappa)$ denoting the discrete space on a set of cardinality $\kappa$.)

(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?

@Paul: The only reason I was certain the question had been asked before was because I had answered it; then I had to find it.

1 Answer