Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2008 / A / 1 / d / Solution

The lower limit topology on the real line is the lower limit topology. Every nonempty basic interval $[a,b)$ contains a rational number, so the rational numbers are a countable dense subset. By the preceding product argument, ${\mathbb{Q}}^2$ is dense in the square, proving its topological separability.

Consider the uncountable anti-diagonalIt is a discrete subset: the basic rectangle $[t,t+ϵ)\times [-t,-t+ϵ)$ intersects $D$ only at $(t,-t)$, because both coordinates have increased weakly and their sum can remain zero only if neither increases.

It is also closed. At a point $(u,v)$ off the anti-diagonal, $u+v\ne 0$. If this sum is positive, every sufficiently small lower-limit rectangle based at $(u,v)$ still has positive coordinate sum. If it is negative, choose $ϵ>0$ with $2ϵ<-(u+v)$; then $[u,u+ϵ)\times [v,v+ϵ)$ has negative coordinate sum throughout. These neighborhoods avoid $D$, so its complement is open.

If the square were Lindelöf, its closed discrete subset $D$ would be countable by part (c), contradicting its bijection with $\mathbb{R}$. Hence