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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Union of Upper Limit Topology and Lower Limit Topology is Discrete Topology

The basis for upper and lower limit topology is $\lbrace (a,b]\mid a<b\rbrace$ and $\lbrace [a,b)\mid a<b\rbrace$ respectively. But the basis of discrete topology on $\mathbb{R}$ is $\lbrace\...
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1answer
39 views

Is Sorgenfrey topology generated by interval $[a,b)$ with a being irrational, $b$ being rational

So the aim is to show that for any open set of Sorgenfrey topology, such open set can be expressed as any union of $[a,b)$, given a being irrational and b rational. How do we show this? or disprove ...
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51 views

What are the path components of $\mathbb{R}_l$?

I am trying to solve What are the path components of $\mathbb{R}_l$? We know that $\mathbb{R}_l$ is lower limit topology. Definition of path component here I am thinking that the path components of $...
2
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1answer
56 views

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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1answer
50 views

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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39 views

Sorgenfreyline proof or disproof [closed]

$(\Bbb{R},\tau)$ is Sorgenfrey topological space, also known as lower limit topology, $x \in \Bbb{R}$ and $\mathscr{N}(x)$ shows all neighboorhoods of $x$. Prove or disprove $$\bigcap_{U \in \mathscr{...
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1answer
58 views

Proof Verification: $\mathbb{R}_l$ is Lindelöf.

My attempt: It will suffice to show that every open covering of $\mathbb{R}_l$ by basic elements contains a countable subcollection covering $\mathbb{R}_l$. Let $A = \{[a_{\alpha}, b_{\alpha} ) \}_{\...
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1answer
99 views

Why is the topology on the Sorgenfrey line not second countable?

For context, let me clarify some things. Our set is $\mathbb{R}$. The topology on $\mathbb{R}$ is the topology generated by the arbitrary unions of closed-open intervals $[a,b)$ with $a,b \in \mathbb{...
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1answer
26 views

Verifying if it is a legitimate basis for a topology

I have been given the task of showing that $\mathcal{B}:=\{[a,b)\vert -\infty<a,b<\infty\}$ is a basis for a topology on $\mathbb{R}$. My problem with it is that there is not a finite union of ...
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2answers
63 views

Is the set {$1-\frac{1}{n} | n\in \mathbb N$} U {1} compact under Sorgenfrey?

I am being asked if the set {$1-\frac{1}{n} | n\in \mathbb N$} $\cup$ {1} is compact under Sorgenfrey topology. I believe it is not, because we can take the open cover K = [1,2)$\cup$[0,$1- \frac{1}{1}...
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3answers
59 views

If f: $\mathbb{R}_{l} \rightarrow S_{\Omega}$ is continuous, then f is not injective.

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, $\...
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1answer
177 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 ...
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2answers
33 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 $\...
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1answer
60 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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1answer
28 views

Quotient map from Sorgenfrey plane to Real Line

I have been told that $\alpha: \mathbb{R}_\ell^2 \longrightarrow \mathbb{R} $, defined by $\alpha(x,y) = x+y$ is a quotient map. From the definition in Munkres, it needs to be the case that $U \subset ...
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61 views

Prove that a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous in lower and upper limit topologies is constant.

Let $f: \mathbb{R} \to \mathbb{R}$ a function that is continuous in the topologies $\mathcal{T}_{u} \to \mathcal{T}_{[,)}$ and $\mathcal{T}_{u} \to \mathcal{T}_{(,]}$, where $\mathcal{T}_{u}$ is the ...
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60 views

Examples of topologies in which all open sets are countable union of base elements, but the topology itself doesn't have countable bases.

In second countable topology(i.e. has countable bases), obviously all open sets are countable union of base elements. But the inverse seems not true. The sorgenfrey line $R_l$ seems to be an example. ...
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1answer
24 views

IVT-related Topology question

Is $[a,b]$ in ${\mathbb{R}_L}$ connected in the subspace topology? I am trying to see whether or not the IVT applies for $[a,b]$ in the topology inherited from $\mathbb{R}_L$ instead of $[a,b]$ ...
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4answers
207 views

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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92 views

Topologies on $\mathbb{R}$

I have found this in the internet: Suppose$$\mathcal B=\{[a,b)\mid a\in\Bbb R\wedge b\in\Bbb Q\wedge a<b\},$$ $$\mathcal B_1=\{[a,b)\mid a\in\Bbb R\wedge b\in\Bbb R\wedge a<b\},$$and$$\mathcal ...
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429 views

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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1answer
47 views

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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2answers
96 views

Lower limit topology on $\mathbb R$ is regular

I want to prove that lower limit topology $\Bbb R_l$ is regular and am taking the approach as follows: Let $A$ be a closed set and $x$ a point in $\Bbb R_l$ such that $x \notin A$. Let $a \in A$. If $...
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31 views

Example 4, Sec. 30, and Example 3, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: How is this set closed in $\mathbb{R}_l^2$?

Let $\mathbb{R}_l$ denote the set of real numbers with the lower limit topology having as a basis the collection of all the closed-open intervals of the form $[a, b)$, where $a$ and $b$ are any real ...
2
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2answers
211 views

Proving $R_L \times R_L$ is completely regular. Meaning $R_L \times R_L$ is an example of a space which is completely regular, but not normal

Can I please receive help/feedback on my proof for the problem below? Thank you $\def\R{{\mathbb R}}$ Prove that $\R_L \times \R_L$ is completely regular. This means $\R_L \times \R_L$ is an example ...
2
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1answer
294 views

Questions about completely normal spaces.

I'm trying to solve the next problem: A topological space $(X,\tau)$ is called completely normal if, and only if, every subspace is normal. Prove that the following conditions are equivalent: a) $X$ ...
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29 views

Sorgenfrey-Line, [x, ->) represented as a union of half-intervals

Given that $x \in R:$ $$[x,\rightarrow ) := \{y \in R\ | \ y \ge x \} = \bigcup \{[x, x + n) \ | \ n \in N\} $$ What I don't understand here is why the right equality holds (from which will follow ...
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1answer
102 views

$\epsilon - \delta$ definition of continuity

I'm trying to give an $\epsilon - \delta$ definition for a function $f: \mathbb{R}_l \rightarrow \mathbb{R}$ to be continuous, where $\mathbb{R}_l$ denote $\mathbb{R}$ with lower limit topology. Here ...
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2answers
28 views

If $f:X\to Y$ is defined by $f(x)=1,x<0$; $f(x)=2,x\geq 0$, and $X$ and $Y$ have Sorgenfrey topology, is $f$ continuous?

Let $X$ and $Y$ be spaces. If $f:X\to Y$ is defined by $f(x)=1,x<0$; $f(x)=2,x\geq 0$, and $X$ and $Y$ have Sorgenfrey topology, is $f$ continuous? I am not sure how to think about this. Any ideas?...
4
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1answer
52 views

Example 2, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Normality of $\mathbb{R}_l$ --- Why are these two sets disjoint?

The set $\mathbb{R}$ of real numbers with the lower limit topology having as a basis the collection of all closed-open intervals $[a, b)$, where $a, b \in \mathbb{R}$ with $a < b$, is denoted by $\...
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1answer
360 views

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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37 views

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 ...
2
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1answer
141 views

$\mathscr{B} = \{ [a, b) | a< b \in \mathbb{R} \}$ is a basis for a Topology in $\mathbb{R}$

I just want to ask if my proof for this problem is correct. $$\mathscr{B} = \{ [a, b) | a< b \in \mathbb{R} \}$$ is a basis for a Topology in $\mathbb{R}$ . Here is my proof: Let $x \in \mathbb{...
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2answers
97 views

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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1answer
285 views

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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0answers
57 views

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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2answers
224 views

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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1answer
203 views

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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1answer
105 views

Sorgenfrey Line is not a topological vector space

Show that Sorgenfrey line is not a topological vector space. My attempt: We know that if $X$ is a topological vector space, then the map $$x\mapsto \alpha x$$ is a homeomorphism for each scalar $\...
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2answers
3k views

$\mathbb R_l$ 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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533 views

Show that a subset of $\mathbb{R}$ is compact in upper limit topology

I want to show that $A = [0,1]$ is not a compact subspace of $\mathbb{R}$, where $\mathbb{R}$ has the upper limit topology with open sets of the form $(a,b] = \{x \in \mathbb{R}\space|\space a < x \...
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0answers
278 views

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 $\tau$?...
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3answers
2k 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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1answer
336 views

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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2answers
2k views

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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1answer
1k views

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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2answers
89 views

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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2answers
2k views

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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2answers
939 views

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 ...
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1answer
929 views

Regular topological spaces need not to be normal

I was looking for a counterexample for the following statement: "A regular topological space need not to be normal." I don't understand how to use the lemma to prove Theorem 7: http://fac.hsu.edu/...