Questions tagged [sorgenfrey-line]
For questions about the Sorgenfrey line ($\mathbb{R}$ with the lower limit topology) and closely related spaces.
86
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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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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 ...
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2
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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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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 ...
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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 $$\cup_\...
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Union of Upper Limit Topology and Lower Limit Topology is Discrete Topology
The basis for upper and lower limit topology is $\lbrace (a,b]\mid a<b\rbrace$ and $\lbrace [a,b)\mid a<b\rbrace$ respectively. But the basis of discrete topology on $\mathbb{R}$ is $\lbrace\...
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Is Sorgenfrey topology generated by interval $[a,b)$ with a being irrational, $b$ being rational
So the aim is to show that for any open set of Sorgenfrey topology, such open set can be expressed as any union of $[a,b)$, given a being irrational and b rational.
How do we show this? or disprove ...
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What are the path components of $\mathbb{R}_l$?
I am trying to solve
What are the path components of $\mathbb{R}_l$?
We know that $\mathbb{R}_l$ is lower limit topology.
Definition of path component here
I am thinking that the path components of $...
user886636
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Describe the topology $L$ inherits as a subspace of $\mathbb{R}_l \times \mathbb{R}$
$\textbf{Notations:}$ $L$ is a line in plane and $\mathbb{R}_l$ is the lower limit topology.
$\textbf{Question:}$ Describe the topology $L$ inherits as a subspace of $\mathbb{R}_l \times \mathbb{R}$.
...
user886636
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What are the continuous functions $\mathbb{R} \rightarrow \mathbb{R}_l$? [duplicate]
Consider $\mathbb{R}_l$ be the lower limit topology which consists basis element of the form $[a,b)$ where $a<b$ and $a,b \in \mathbb{R}$.
I am facing trouble to solve what are the continuous ...
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Sorgenfreyline proof or disproof [closed]
$(\Bbb{R},\tau)$ is Sorgenfrey topological space, also known as lower limit topology, $x \in \Bbb{R}$ and $\mathscr{N}(x)$ shows all neighboorhoods of $x$. Prove or disprove $$\bigcap_{U \in \mathscr{...
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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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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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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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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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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, $\...
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Why is the Sorgenfrey Line not second countable?
A topology is second countable if there exist a countable basis. For the Sorgenfrey line, each basis are of the form [a,b) where a and b are both real numbers. This is uncountable, but if I restrict ...
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Show each interval (a,b) is open in the basis B
I am using Topology Without Tears by Morris
Let B={[a,b)$\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 $\...
user830852
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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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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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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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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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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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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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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$$\mathcal ...
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Continuous function from $\mathbb{R}$ standard topology to $\mathbb{R}_l$ lower limit topology
The 2nd chapter of Topology by Munkres discussed the identity function
$$
f: \mathbb{R} \rightarrow \mathbb{R}_l
$$
from $\mathbb{R}$ (with standard topology) to $\mathbb{R}_l$ (with lower limit ...
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Validation of my proof for proving that the Sorgenfrey Line does not satisfies the second axiom of countability
In an exercice I am asked to prove the following:
Prove that the Sorgenfrey Line does not satisfies the second axiom of contability.
This is my second proof for this exercise because the first one ...
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Lower limit topology on $\mathbb R$ is regular
I want to prove that lower limit topology $\Bbb R_l$ is regular and am taking the approach as follows:
Let $A$ be a closed set and $x$ a point in $\Bbb R_l$ such that $x \notin A$.
Let $a \in A$. If $...
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Example 4, Sec. 30, and Example 3, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: How is this set closed in $\mathbb{R}_l^2$?
Let $\mathbb{R}_l$ denote the set of real numbers with the lower limit topology having as a basis the collection of all the closed-open intervals of the form $[a, b)$, where $a$ and $b$ are any real ...
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Proving $R_L \times R_L$ is completely regular. Meaning $R_L \times R_L$ is an example of a space which is completely regular, but not normal
Can I please receive help/feedback on my proof for the problem below? Thank you
$\def\R{{\mathbb R}}$
Prove that $\R_L \times \R_L$ is completely regular. This means $\R_L \times \R_L$ is an example ...
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Questions about completely normal spaces.
I'm trying to solve the next problem: A topological space $(X,\tau)$ is called completely normal if, and only if, every subspace is normal. Prove that the following conditions are equivalent:
a) $X$ ...
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Sorgenfrey-Line, [x, ->) represented as a union of half-intervals
Given that $x \in R:$
$$[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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$\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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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?...
user482939
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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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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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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 ...
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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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$\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} \}$$ is a basis for a Topology in $\mathbb{R}$ .
Here is my proof:
Let $x \in \mathbb{...
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Lower limit topology and empty set
How is the empty set generated by the arbitrary union of half open intervals of the form $[a,b), a<b, a,b\in R$.
I can't come up with a union.
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Connected topologies on $\mathbb{R}$ strictly between the usual topology and the lower-limit topology
It is well-known that the usual order/metric topology on $\mathbb{R}$ is connected, and the lower-limit topology is not connected (it is even totally disconnected). We also know that the lower-limit ...
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One-Sided Notion of Topological Closure
Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this:
Let $A$ be a subspace of $\mathbb{R}$. Define an operation ...
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Disconnectedness of closed intervals in Sorgenfrey's line
In order to prove Sorgenfrey's line is totally disconnected I took the long road and proved every type of subset except singletones (intervals and rays) is disconnected. Everyone except for closed ...
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Certain Subset of Sorgenfrey Plane is Closed
Note that $L = \{(x,-x) \mid x \in \Bbb{R} \}$ is closed. Then if $A$ is closed in $L$, then it will also be closed in $\Bbb{R}^2_\ell$. According to Munkres, $L-A$ will also be closed, but I am ...
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Sorgenfrey Line is not a topological vector space
Show that Sorgenfrey line is not a topological vector space.
My attempt:
We know that if $X$ is a topological vector space, then the map $$x\mapsto \alpha x$$ is a homeomorphism for each scalar $\...
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$\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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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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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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Why the Sorgenfrey plane is not Lindelöf?
Definition: A space for which every open covering contains a countable subcovering is called a Lindelöf space
In the book Topology written by Munkres it is said that the Sorgenfrey plane is not ...
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