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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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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 ...
Raven's user avatar
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-1 votes
2 answers
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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 ...
Grimm Troupe's user avatar
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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 ...
Paul's user avatar
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Consider the space $\Bbb R_l$ that is $\Bbb R$ with the lower limit topology. Show that $\Bbb R_l$ is Lindelöf. [duplicate]

Consider the space $\Bbb R_l$ that is $\Bbb R$ with the lower limit topology. Show that $\Bbb R_l$ is Lindelöf. I am trying to understand a proof, but I don't know why the result implies that $\Bbb ...
Walker's user avatar
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2 votes
2 answers
107 views

Sorgenfrey line - what is wrong with this argument

Consider $X$ the reals with Euclidean topology, $Y$ the reals with Sorgenfrey topology, $f: X \rightarrow Y$ identity mapping $f(x) = x$. I know that this mapping is not continuous, because $U = [0,1)$...
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4 votes
1 answer
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Total separatedness and separation axioms

Recall that a nonempty topological space $X$ is said to be totally separated iff, for every distinct points $x,y \in X$, there is a separation $U,V$ of $X$ such that $x \in U$ and $y \in V$. It can be ...
Dannyu NDos's user avatar
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Arbitrary union in lower limit topology

So the lower limit topology on R has open sets of the form [a, b), where a<b, so far so good, my question is, if the arbitrary union of left closed sets can be open, that is for example $$\cup_\...
Jocsan Ariel Hernandez Barahon's user avatar
2 votes
1 answer
456 views

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\...
mali1234's user avatar
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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 ...
youngeAn's user avatar
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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 $...
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2 votes
1 answer
155 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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1 answer
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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 ...
user092022's user avatar
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1 answer
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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{...
nazizede's user avatar
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1 answer
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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} ) \}_{\...
Hoang Nguyen's user avatar
5 votes
1 answer
333 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{...
SunRoad2's user avatar
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1 answer
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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 ...
Novice Sprinter's user avatar
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2 answers
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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}...
UlisesQLL's user avatar
3 votes
3 answers
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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, $\...
george45's user avatar
1 vote
1 answer
661 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 ...
Bill's user avatar
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2 answers
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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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3 votes
1 answer
114 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 ...
flower137's user avatar
0 votes
1 answer
55 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 ...
Fishes_'s user avatar
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1 answer
335 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 ...
gal16's user avatar
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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. ...
Alex Fan's user avatar
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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 $[a,b]$ ...
nate_hudnall's user avatar
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4 answers
405 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 ...
PatrickR's user avatar
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0 votes
2 answers
138 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 ...
Averroes2's user avatar
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2 answers
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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 ...
whitegreen's user avatar
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1 answer
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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 ...
Eduardo Magalhães's user avatar
2 votes
2 answers
448 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 $...
Singh_Gunjeet's user avatar
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0 answers
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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 ...
Saaqib Mahmood's user avatar
2 votes
2 answers
630 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 ...
rudinsimons12's user avatar
2 votes
1 answer
497 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) $X$ ...
Dendrilo's user avatar
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0 answers
34 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 ...
Aelx's user avatar
  • 481
1 vote
1 answer
130 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 ...
nico23's user avatar
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0 votes
2 answers
30 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?...
user avatar
4 votes
1 answer
75 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 $\...
Saaqib Mahmood's user avatar
3 votes
1 answer
736 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 $\tau \subsetneq \tau '$. ...
Kumar's user avatar
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1 vote
1 answer
3k views

How to show $R_l$ is Lindelöf space? [duplicate]

I wanted to prove following exercise $R_l$ lower limit topology is Lindelöf Space. Lindelöf space is space with every cover has countable cover. I tried but I am not able to even start. Please give me ...
Curious student's user avatar
0 votes
2 answers
45 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 ...
Aitor Vazquez's user avatar
2 votes
1 answer
226 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{...
Mashed Potato's user avatar
2 votes
2 answers
191 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.
Jhon Doe's user avatar
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4 votes
1 answer
553 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 ...
stochastic randomness's user avatar
2 votes
0 answers
66 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 ...
John Samples's user avatar
0 votes
2 answers
327 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 ...
Friedman's user avatar
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2 votes
1 answer
400 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 ...
user193319's user avatar
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4 votes
1 answer
138 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 $\...
Sahiba Arora's user avatar
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0 votes
2 answers
5k 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?
Gauss's user avatar
  • 21
1 vote
3 answers
803 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 \...
Enrico's user avatar
  • 15
0 votes
0 answers
343 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 $\tau$?...
kasp9201's user avatar
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