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# 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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### 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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If f: $\mathbb{R}_{l} \rightarrow S_{\Omega}$ is continuous, then f is not injective. I've been trying to solve this problem for a few days, but I haven't been able to see how can I do it. First, $\... 0 votes 1 answer 400 views ### Why is the Sorgenfrey Line not second countable? A topology is second countable if there exist a countable basis. For the Sorgenfrey line, each basis are of the form [a,b) where a and b are both real numbers. This is uncountable, but if I restrict ... 0 votes 2 answers 56 views ### Show each interval (a,b) is open in the basis B I am using Topology Without Tears by Morris Let B={[a,b)$\in R$:a<b} B is a basis for some topology$\tau$on R,but not the Euclidean one. Regardless show for each interval (a,b)is open in (R$\... 91 views

### 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?
1 vote