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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Describe the topology $L$ inherits as a subspace of $\mathbb{R}_l \times \mathbb{R}$

$\textbf{Notations:}$ $L$ is a line in plane and $\mathbb{R}_l$ is the lower limit topology. $\textbf{Question:}$ Describe the topology $L$ inherits as a subspace of $\mathbb{R}_l \times \mathbb{R}$. ...
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What are the continuous functions $\mathbb{R} \rightarrow \mathbb{R}_l$? [duplicate]

Consider $\mathbb{R}_l$ be the lower limit topology which consists basis element of the form $[a,b)$ where $a<b$ and $a,b \in \mathbb{R}$. I am facing trouble to solve what are the continuous ...
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Continuous function from $\mathbb{R}$ standard topology to $\mathbb{R}_l$ lower limit topology

The 2nd chapter of Topology by Munkres discussed the identity function $$f: \mathbb{R} \rightarrow \mathbb{R}_l$$ from $\mathbb{R}$ (with standard topology) to $\mathbb{R}_l$ (with lower limit ...
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Validation of my proof for proving that the Sorgenfrey Line does not satisfies the second axiom of countability

In an exercice I am asked to prove the following: Prove that the Sorgenfrey Line does not satisfies the second axiom of contability. This is my second proof for this exercise because the first one ...
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Topology Munkres ($2^\text{ed}$) $\S 16$ Exercise $2$: Subspaces of Finer Topologies

The following theorems are well known to me: (i) Suppose $\tau$ and $\tau '$ are two topologies on a given set $X$. Then, $\tau '$ is said to be strictly finer than $\tau$ if $\tau \subsetneq \tau '$. ...
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Compact sets through topologies over a set

I have to show that a set is compact on the Sorgenfrey line. I can prove that it is indeed compact on the usual topology over the real line and my idea is to say that, since the Sorgenfrey topology is ...
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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?