# 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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### Real line and the Sorgenfrey line have the same dense subsets

I am trying to prove that the real line and the Sorgenfrey line have the same dense subsets. That is, $A\subset \Bbb R$ is dense under the lower limit topology if and only if it is dense under the ...
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### how to prove that the topology generated by the left-closed intervals is finer than the usual topology

The idea is to prove that the open intervals (like $]a,b[$) are contained in the topology of the left-closed sets ($[a,b[$), but I cannot see a way of generating open sets from half-closed sets. (Same ...
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### $\mathbb{R}_{\rm Sorg.}$ is paracompact but $\mathbb{R}_{\rm Sorg.} \times \mathbb{R}_{\rm Sorg.}$ is not

I’m trying to solve this problem: $\def\bbR{\mathbb{R}} \def\RSorg{\bbR_{\rm Sorg.}} \def\calR{\mathcal{R}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \DeclareMathOperator{\range}{range}$ Prove ...
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### Are $[0, 1)$ and $[0, 1]$ homeomorphic subspaces of the Sorgenfrey line?

My argument is that $\{1\}$ is a connected clopen subspace of $[0, 1]$ while the only connected subspaces of $[0, 1)$ are singular sets, which are not open in $[0, 1)$, so the spaces must not be ...
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### $\mathbb R_l$ is not connected.

How to show $\mathbb R_l$ (lower limit topology on $\mathbb R$) is not connected?Means how any basis element of $\mathbb R_l$ can be written as the union of two separated sets?
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