# 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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### Lower limit topology and empty set

How is the empty set generated by the arbitrary union of half open intervals of the form $[a,b), a<b, a,b\in R$. I can't come up with a union.
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### Connected topologies on $\mathbb{R}$ strictly between the usual topology and the lower-limit topology

It is well-known that the usual order/metric topology on $\mathbb{R}$ is connected, and the lower-limit topology is not connected (it is even totally disconnected). We also know that the lower-limit ...
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### One-Sided Notion of Topological Closure

Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this: Let $A$ be a subspace of $\mathbb{R}$. Define an operation ...
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### Disconnectedness of closed intervals in Sorgenfrey's line

In order to prove Sorgenfrey's line is totally disconnected I took the long road and proved every type of subset except singletones (intervals and rays) is disconnected. Everyone except for closed ...
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### Certain Subset of Sorgenfrey Plane is Closed

Note that $L = \{(x,-x) \mid x \in \Bbb{R} \}$ is closed. Then if $A$ is closed in $L$, then it will also be closed in $\Bbb{R}^2_\ell$. According to Munkres, $L-A$ will also be closed, but I am ...
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### Boundary points with a lower limit topology

Let $\tau$ be a lower limit topology (also called the Sorgenfrey topology) on $\mathbb{R}$. If $a<b$, then for an interval $A=[a,b)$ on the real number line what is the boundary points w.r.t $\tau$?...
632 views

### $\mathbb{R}_\ell$ is not locally compact

Consider $\mathbb{R}_\ell$ be the the 'Sorgenfrey line': Real line with the topology constructed from the intervals $\{[a,b):a<b\}$. Prove that $\mathbb{R}_\ell$ is not locally compact.
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### Let $X=\mathbb{R}$ with the lower limit topology, and $Y=\mathbb{R}$ with the upper limit topology. Is $[1,2) \times [1,2)$ open in $X \times Y$

I don't believe $[1,2)$ is open in Y, so the product topology, $X\times Y$, is then not open. As I'm reading Topology Without Tears, I see Proposition 8.1.4 that discusses the product space being ...
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### Prove that the Sorgenfrey line is totally disconnected

Problem: Let $\mathbb{R}_l$ denote the topological space whose underlying set is the real line $\mathbb{R}$ and the topology is generated by the half closed intervals $[a,b)$. Prove that the ...
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### Prove that the Sorgenfrey line is not connected

Problem: Let $\mathbb{R}_l$ denote the topological space whose underlying set is the real line $\mathbb{R}$ and the topology is generated by the half closed intervals $[a,b)$. Prove that the ...
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### Example of cont. map. $\psi: \mathbb{R}_l \to \mathbb{R}_l$ such that $\phi: \mathbb{R}\to \mathbb{R}$, def. by $\phi(x)=\psi(x)$, is not cont.

$\mathbb{R}_l$ denotes the Sorgenfrey line or the Lower limit topology generated by the half-open intervals $[a,b)$ and $\mathbb{R}$ denotes the usual euclidean topology in $\mathbb{R}$. Can ...
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### Interval $[0,1]$ is neither compact nor connected in the Sorgenfrey line.

Let $A=[0,1]$. Show that $A$ is neither compact nor connected in the Sorgenfrey line, $\tau_{[,)}$, and that there is no neighborhood of $0$ compact. For the connectedness part, I thought that $[0,1)$...
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### Open sets which are not closed in the Sorgenfrey line

Basically, it is a simple fact about the Sorgenfrey line that: the only connected sets are the singelton sets. the open set in Sorgenfrey line $(b,\infty)$ is not closed. But are there other open ...