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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4answers
71 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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3answers
73 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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2answers
54 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
9 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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0answers
28 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 ...
1
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1answer
124 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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0answers
25 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
73 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
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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?...
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1answer
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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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1answer
194 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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2answers
28 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
85 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{...
2
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2answers
71 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.
4
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1answer
124 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
56 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
124 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 ...
2
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1answer
148 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 ...
3
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1answer
91 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
2k 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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3answers
305 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
202 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
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$\mathbb{R}_\ell$ 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
267 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
1k 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 ...
3
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1answer
730 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
77 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
931 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
657 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
604 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/...
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1answer
64 views

Is the given function $f$ continuous?

Problem Let $\mathbb{R}_l$ denote the reals with lower limit topology, and let $\mathbb{R}_l\times \mathbb{R}_l$ have the product topology. Then the map $f:\mathbb{R}_l\times\mathbb{R}_l\to\mathbb{R}...
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1answer
963 views

Compact sets of lower limit topology

Find all the compact sets of the lower limit topology in $\mathbb{R}$, i.e. the topology given by basis $\{[a,b)|a,b \in \mathbb{R} \}$. What I have got so far: necessary conditions: the compact ...
3
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1answer
164 views

Sorgenfrey line is not orderable [closed]

How can I prove that the Sorgenfrey line is not a linearly ordered topological space, i.e., that the Sorgenfrey topology on $\mathbb R$ is not the order topology for some linear ordering? I tried to ...
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1answer
81 views

How to prove Sorgenfrey topology stronger than standard?

I understand that my title's statement is true and I'm not sure why. Any [a,b) set is open on Sorgenfrey's so I guess I could use a union of all [a + 1/n, b) sets in order to get the (a,b) set which ...
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2answers
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Cheap proof that the Sorgenfrey line is normal?

It is very easy to prove that the Sorgenfrey line is completely regular: To separate a point $x$ from a closed set $F$, note that there is an interval $[x,y)$ disjoint from $F$ and observe that the ...
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1answer
1k views

topology (upper limit and lower limit)

I have to show that upper limit topology and lower limit topology on $\mathbb{R}$ (Real line) are not comparable. But suppose if we take $[a,b)$ and $(a-1,b]$, where $a-1 > a$, then isn't it ...
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2answers
132 views

Why is $\mathfrak{c}$ the cardinality of the lower limit topology on $\mathbb{R}$?

Why is $\mathfrak{c} = |\mathbb R|$ the cardinality of the lower limit topology on $\mathbb{R}$? An open set in the lower limit topology is of the form $[a,b)$. I can clearly see why the cardinality ...
2
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1answer
131 views

Prob. 2, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: Compactness of $[0,1]$ in the lower limit topology

Let $\mathbb{R}_l$ denote the set of real numbers with the topology having as a basis all the half open intervals $[a,b)$ on the real line. Then is the closed interval $[0,1]$ compact as a subspace ...
4
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1answer
1k views

Lower Limit Topology Properties

I am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of ...
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2answers
3k views

The closure of $(0,1)$ in the lower-limit topology on $\mathbb{R}$

I've been given the topology $\tau_l$ on $\mathbb{R}$ generated by the subbasis consisting of all half open intervals $[a,b)$. I've concluded therefore that one can define the topology as: $$\tau_l :=...
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1answer
2k views

Is the Sorgenfrey Line second countable? [duplicate]

The Sorgenfrey topology on $\mathbb{R}$ is the topology whose basic open sets are of the form $[a,b)$ where $a < b \in \mathbb{R}$. Does it have a countable base? (I suspect not.) Certainly it is ...
27
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2answers
9k views

$\mathbb{R}$ with the lower limit topology is not second-countable

I am trying to prove that $\mathbb{R}$ with the lower limit topology is not second-countable. To do this, I'm trying to form an uncountable union $A$ of disjoint, half-open intervals of the form $[a, ...
2
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1answer
229 views

$\{1/n\}_n$ converges in the Sorgenfrey line

The following is my attempted proof that the sequence $\{ \frac{1}{n} \}_n$ converges in the Sorgenfrey line. Offer criticism, please! Consider $\{\frac{1}{n}\}_n=(1,\frac{1}{2},\frac{1}{3},\cdots)$...
3
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1answer
151 views

$\{1-\frac{1}{n} \}_n$ does not converge in the Sorgenfry line

I am trying to prove that the sequence $\{1-\frac{1}{n} \}_n$ does not converge in the Sorgenfry line. Below is my attempt. Consider $\{1-\frac{1}{n}\}_n=(0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots)...
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2answers
223 views

Continuity of addition under the lower limit topology

Letting $\Bbb R_\ell$ denote $\mathbb{R}$ with the lower limit topology, prove that the function $f : \Bbb R_\ell \times \Bbb R_\ell \to \Bbb R_\ell$ defined by $f((x,y)) = x+y$ is continuous.
3
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1answer
643 views

Show that the Sorgenfrey line does not have a countable basis.

I am trying to understand this proof from Munkres' book which shows that the Sorgenfrey line does not have a countable basis. His proof is: Let $\beta$ be a basis for $\mathbb{R}_l$. Choose for each $...
6
votes
2answers
5k views

Is the lower limit topology finer than the standard topology on $\mathbb{R}$?

Is the lower limit topology finer than the standard topology on $\mathbb{R}$? In Lemma 13.4 on p.82 of Munkres' Topology (2nd ed.), it is stated that the lower limit topology is (strictly) finer ...
2
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1answer
2k views

Is set on lower-limit topology path-connected?

Is $\mathbb R$ endowed with the left-hand topology (also called lower limit topology) path-connected? Intuitively, I know that the answer is yes but I'm not sure how to prove it. Would it suffice to ...
4
votes
2answers
1k views

Every subspace of $\mathbb{R}$ with the lower limit topology is separable

Let us consider the lower limit topology $τ=\{G⊂R: (∀x∈G)(∃ϵ>0)([x,x+ϵ)⊂G)\}$ on $\mathbb{R}$. I am trying to show that any subspace of $(\mathbb{R},τ)$ is separable, but couldn't find the ...
3
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1answer
574 views

$\mathbb{R}_{S}$ = (Sorgenfrey line) is a Baire Space

I'm tried to prove that $\mathbb{R}_{S}$ = (Sorgenfrey line) is a Baire Space. I find that my prove is correct, but I'm not sure. $\{U_n; n \in \mathbb{N}\}$ are a collection of open and dense sets ...