# Questions tagged [sorgenfrey-line]

For questions about the Sorgenfrey line ($\mathbb{R}$ with the lower limit topology) and closely related spaces.

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### The Sorgenfrey plane and the Niemytzki plane are Baire spaces

A space $X$ is called a Baire space if every countable intersection of open dense sets is dense. By the Baire category theorem, every complete metric space is Baire and every locally compact ...
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### Is the Sorgenfrey Line second countable? [duplicate]

The Sorgenfrey topology on $\mathbb{R}$ is the topology whose basic open sets are of the form $[a,b)$ where $a < b \in \mathbb{R}$. Does it have a countable base? (I suspect not.) Certainly it is ...
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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 $[a, ... 1answer 229 views ###$\{1/n\}_n$converges in the Sorgenfrey line The following is my attempted proof that the sequence$\{ \frac{1}{n} \}_n$converges in the Sorgenfrey line. Offer criticism, please! Consider$\{\frac{1}{n}\}_n=(1,\frac{1}{2},\frac{1}{3},\cdots)$... 1answer 151 views ###$\{1-\frac{1}{n} \}_n$does not converge in the Sorgenfry line I am trying to prove that the sequence$\{1-\frac{1}{n} \}_n$does not converge in the Sorgenfry line. Below is my attempt. Consider$\{1-\frac{1}{n}\}_n=(0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots)...
Letting $\Bbb R_\ell$ denote $\mathbb{R}$ with the lower limit topology, prove that the function $f : \Bbb R_\ell \times \Bbb R_\ell \to \Bbb R_\ell$ defined by $f((x,y)) = x+y$ is continuous.