It is very easy to prove that the Sorgenfrey line is completely regular: To separate a point $x$ from a closed set $F$, note that there is an interval $[x,y)$ disjoint from $F$ and observe that the characteristic function of $[x,y)$ is continuous because the half-open intervals generating the topology are clopen. The proof of […]

Let $X=[0,1)\times[0,1)$, $\tau$ its topology with base $$\beta = \{ [a,b)\times[c,d): 0 \leq a < b \leq 1, 0 \leq c < d \leq 1 \}\;.$$ Please help me prove, that it is regular, but not a normal topological space.

I am trying to prove that $\mathbb{R}$ with the lower limit topology is not second-countable. To do this, I’m trying to form an uncountable union $A$ of disjoint, half-open intervals of the form $[a, b)$, $a < b$. Is this possible? I think this would imply the $A$ is open but no countable union of […]

How can one show that the Sorgenfrey line is hereditarily Lindelöf (that is, all subspaces of the Sorgenfrey line are Lindelöf)? I know the Sorgenfrey line is Lindelöf and hence every closed subspace is Lindelöf.

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