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Questions tagged [first-countable]

For questions about first countable topological spaces, i.e., space with countable local base at each point.

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countable set of subsets of $\mathbb{Z}^{d}$ [closed]

Is the set of finite subsets of $\mathbb{Z}^d$ which contain a prescribed vertex and are compact and connected, countable? Hint: It is clear that it is not countable without the restriction that the ...
QuantumLogarithm's user avatar
2 votes
1 answer
156 views

Definition of compactness in terms of convergent sequences.

This is a question about first countable spaces. Topology of such spaces can be defined in terms of convergent sequences, and many topological properties of such spaces can be expressed in terms of ...
Ilia's user avatar
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If a topological space X is not first countable, is it second countable?

I know that topological space is first countable if each point has a countable neighborhood basis. A neighborhood basis at a point. Consider the following topological space, $X=R$ with $\mathcal T=\{A\...
Anonymous's user avatar
2 votes
2 answers
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Are dual spaces to separable normed spaces first-countable?

In [1] I found the following theorem (roughly translated by me) Satz 13.10 If $X$ is a separable normed $k$-vector space ($k = \mathbb R$ or $\mathbb C$) with continuous dual space $X'$, then the ...
red_trumpet's user avatar
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1 answer
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Proving first countability of $\mathbb{R}$ under the particular point topology.

I am trying to show that $\mathbb{R}$ is a first countable space with respect to the particular point topology defined by: $$\mathcal{T} = \{I\subseteq \mathbb{R} : I = \emptyset \text{ or } p\in I\}$$...
omar11235's user avatar
1 vote
3 answers
103 views

Countably close topological spaces

Does every uncountable topological space has a different and mutually "countably close" space? Define "countably close" space $S$ with respect to another space $S'$ both on the ...
Jan Safronov's user avatar
1 vote
0 answers
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Topological spaces with countable open set difference

My question is regarding the existance of "countably close" open sets to those of a parotopological group (or any group equipped with a topology). Consider a group $G$ equipped with a ...
Jan Safronov's user avatar
1 vote
1 answer
102 views

Sequentially Compact implies Compact?

I know that in metric spaces sequentially compact is equivalent to compact. I also know that compact and sequentially compact for general topologies there is no relation between them. But I wanted to ...
Paúl Peñaherrera's user avatar
1 vote
1 answer
104 views

homeomorphism between limit point compact set and 1st countable hausdorff space

X is a limit point compact space and Y is a 1st countable hausdorff space. Then show that bijective, continuous map f:X->Y is a homeomorphosm All we need to show is f is an open or closed map. I've ...
황주영's user avatar
8 votes
2 answers
244 views

Partial limits in general topological spaces

Let $X$ be a general topological space, let $\{x_n\}_{n=1}^\infty\subseteq X$ be a sequence, and let $y\in X$. Suppose that for every $V\subseteq X$ open neighborhood of $y$, the set $\{n\in\mathbb{N}\...
User271828's user avatar
1 vote
0 answers
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Are sober noetherian spaces sequential?

A sequential topological space $X$ has a few different equivalent definitions: $X$ is the quotient of a first-countable space $X$ is the quotient of a metric space Sequentially open subsets of $X$ ...
saolof's user avatar
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Prove that the following properties are all finitely productive

The question goes as follows: Prove that the following properties are all finitely productive (1) $T_0$ and $T_1$ (2) Separable (3) First Countable (4) Second Countable (5) Finite (i.e., the ...
Ryukendo Dey's user avatar
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36 views

First-countable topological spaces

So I have a topology defined as follows: $$ \tau = \{\mathbb{R} \} \cup \{ U \subset \mathbb{R} \ \ | \ \ 0 \notin U \}$$ I have already prooved that is a topology of $\mathbb{R}$ and that the local ...
Juan Otero Rivas's user avatar
1 vote
0 answers
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Countable but not First Countable Space [duplicate]

I came across this question in Chapter 5 of Biglist. I have no idea how to grab a handle on the question. Construct a Topological space $(X, \mathcal{T})$ which is countable (i.e., $X$ is countable) ...
Ryukendo Dey's user avatar
1 vote
3 answers
142 views

Is $\mathbb{R}$ under the countable complement topology path connected? (Proof check)

I'm trying to prove that when $\mathbb{R}$ has the countable complement topology, it is not path connected. I used the following definition of continuous: $f(\overline{A})\subset\overline{f(A)}$. We ...
CuriousCarrot2008's user avatar
1 vote
1 answer
175 views

How First countable topological space implies Fréchet Urysohn space

Here are the definitions: Fréchet-Urysohn space: A topological space $ X $ where for every $ A \subseteq X $ and every $ x \in \text{cl}(A) $, there exists a sequence $ (x_{n})_{n \in \mathbb{N}} $ ...
Infinity's user avatar
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How can the implication metrizable -> first countable be generalized?

I'm making a contribution to the pi-Base to automatically deduce certain spaces are not metrizable based on the lack of first-countability. How can this theorem be generalized to deduce more [non-]...
Steven Clontz's user avatar
2 votes
1 answer
57 views

Extension of non-first countable space

I have the next question: Consider $X$ a topological space that is non-first countable. We can assume that $X$ is Tychonoff. Is there a way to embed $X$ in a topological space $Y$ such that $X$ is ...
Carlos Jiménez's user avatar
1 vote
1 answer
104 views

Topology question on first countable.

This question was asked in the GATE MA 2023 paper: Q.44. Let $(\mathbb{R},\tau)$ be a topological space, where the topology $\tau$ is defined as $$\tau = \{U \subset \mathbb{R}: U = \emptyset \ or \ 1 ...
Gajendra Basavaraju's user avatar
3 votes
1 answer
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Are first and second countability preserved under intersection of topologies?

For a given set $X$ endowed with two topologies $\mathcal{T}$ and $\mathcal{T}'$, i.e. such that $(X,\mathcal{T})$ and $(X,\mathcal{T}')$ are two topological spaces defined on the same $X$, it is easy ...
K. Makabre's user avatar
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1 answer
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Disjoint Union of First Countable Spaces

This question concerns part $d)$ of proposition 3.42 of Lee's book on topological manifolds. Let $ \left( X_{\alpha} \right)_{\alpha \in A} $ be an (arbitrary) indexed family of topological spaces. ...
Nikolawn's user avatar
1 vote
2 answers
57 views

First Countable Spaces and Limit Preserving Functions

I've being struggling with the following problema in Lee's book on topological manifolds. Let $X$ and $Y$ be topological spaces. Let $f:X\to Y$ be a map such that $p_n \to p$ (convergent sequence) in ...
Nikolawn's user avatar
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If $X$ is first countable and $S$ is a subspace of $X$, show that $S$ is first countable

This is part of Lee's Introduction to Topological Manifolds exercise 3.12. While the question contains multiple pieces, I am primarily interested in solving the following: Suppose S is a subspace of ...
WaterDrop's user avatar
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1 answer
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Every first countable countably compact space is sequentially compact

I have already read A first countable, countably compact space is sequentially compact and wish not use limit point compactness in my proof. Please do not reference. Let $X$ be first countable and ...
Grigor Hakobyan's user avatar
1 vote
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103 views

Does $S_{\Omega}$ satisfy the first countability axiom?

While solving an exercise in Munkres' "Topology" (Ex. 12(c) Sect. 24) I have tried proving the following lemma: Let us denote the minimal uncountable well-ordered set by $S_{\Omega}$. Then $...
Matteo Menghini's user avatar
4 votes
1 answer
336 views

Why does satisfying the countability conditions make topological spaces so nice?

Motivation for the question: While I can appreciate the significance of Theorem 30.1 below, and also the Urysohn metrization theorem, I can't seem to understand why either of them is rooted in these ...
tryst with freedom's user avatar
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65 views

In $(X,\tau)$ every sequentially closed set is topologically closed. Does this implies $(X, \tau) $ first countable?

$(X, \tau) $ be a topological space. • $A\subset X$ is said to be closed( or topologically closed) if $X\setminus A\in\tau$ • $A\subset X$ is sequentially closed of for every sequences $(x_n) \subset ...
Ussesjskskns's user avatar
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1 answer
164 views

If $X$ is first-countable then a net converges when a subsequence converges?

Let be $X$ and we assume that $(x_\lambda)_{\lambda\in\Lambda}$ is a net such that there exists a cofinal and increasing map $\varphi$ form $\Bbb N$ to $\Lambda$ such that $\big(x_{\varphi(n)}\big)_{n\...
Antonio Maria Di Mauro's user avatar
1 vote
0 answers
99 views

Exercise 1(a), Section 30 of Munkres’ Topology

(a) A $G_\delta$ set in a space $X$ is a set $A$ that equals a countable intersection of open sets of $X$. Show that in a first-countable $T_1$ space, every one-point set is a $G_\delta$ set. My ...
user264745's user avatar
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1 vote
1 answer
97 views

Definition of Countability in Munkres’ Topology

$X$ have a countable basis at $x$, if $\exists \{U_n \in \mathcal{N}_x |n\in \Bbb{N}\}$ with the following property: $\forall U\in \mathcal{N}_x, \exists m\in \Bbb{N}$ such that $U_m\subseteq U$. If $...
user264745's user avatar
  • 4,173
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0 answers
91 views

Lower semicontinuity of parameter dependent Lebesgue integral

Let $(\Omega,\mathcal F,\mu)$ be a measure space and let $(X,\tau)$ be a topological space with countable base. Suppose we are given a function $f:\Omega \times X \to [0,\infty]$ with the following ...
Alphie's user avatar
  • 4,797
3 votes
2 answers
234 views

$\prod_{n=1}^{\infty}{\mathbb{R}}$ endowed with the box topology is not first countable.

What I'm trying to prove is that if $X^{+}\subseteq X:=\prod_{n=1}^{\infty}{\mathbb{R}}$ is the set of all positive sequences in $\mathbb{R}$, then no sequence of elements in $X^{+}$ converges to the ...
Shuichi's user avatar
  • 431
0 votes
2 answers
117 views

Countable Complement Space is not First-Countable

I am tring to understand the proof given in Countable Complement Space is not First-Countable What I don't understand is that how is it that the intersection of all members of the countable local ...
gbd's user avatar
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0 votes
1 answer
77 views

Topology counterexamples without ordinals

I am looking for three counterexamples in general topology, namely: A set which is sequentially closed, but not closed; A set which is sequentially compact, but not compact; A set which is compact ...
Khalid Wenchao Yjibo's user avatar
0 votes
3 answers
186 views

Countable basis and first countable

A space $X$ is said to have a countable basis at $x$ is there is a countable collection $B$ of neighbourhoods of $x$ such that each neighbourhood of $x$ contains at least one of the most elements from ...
danny's user avatar
  • 906
1 vote
0 answers
55 views

Recasting Algorithmic Information In Terms of Finite Directed _Cyclic_ Graphs?

Any bit-string {0,1}* can be produced by a finite directed cyclic graph, the nodes of which are n-input NOR functions, with at least two arcs directed away from the graph without a terminal connection ...
James Bowery's user avatar
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0 answers
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Why is the ordered square $[0,1]^{2}$ first countable in the dictionary order? [duplicate]

¿Why is the ordered square $[0,1]^{2}$ first countable in the dictionary order? I suppose for any point in $[0,1]^{2}$ I must find a countable basis, but I do not know yet what it should be or how to ...
NoetherNerd's user avatar
2 votes
0 answers
70 views

First countable space and convergent sequences [closed]

Let $(\mathbb{R},T)$ be the co countable topological space where $T=\{A \subseteq \mathbb{R}:A^c \, \text{is countable} \}\cup\{\phi\}$. Take $A=\mathbb{R}/\{1\}$, then $\bar{A}=\mathbb{R}$ The ...
user3911153's user avatar
0 votes
2 answers
100 views

Directly proof $S$ is countable, where $S$ is set of function from $\{0, 1\}$ to $\mathbb{N}$

Suppose $S=\{f_1,f_2,f_3,f_4,f_5,........\}$ where $f_i$ is a function $f:\{0, 1\}\to\mathbb{N}.$ I have to prove $S$ is countable.Then need to prove direct one-to-one correspondence between $S$ and $\...
user avatar
0 votes
1 answer
76 views

Is this a non-first-countable space?

I am trying to prove the topological space $(X, \mathcal T)$ below is not a first-countable space. I tried my best but I'm not sure if it is well proved. Whatever there's any improper detail or it's ...
Wenchuan's user avatar
0 votes
1 answer
61 views

Axioms of countability

Let $\mathbb R$, $T=\{\emptyset\}\cup\{G\subset \mathbb R\mid \mathbb Q\setminus G \text{ is finite}\}$. Let $(\mathbb R, T)$. I have to prove what axioms of countability are verifyed in this space. ...
GoRza's user avatar
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1 vote
0 answers
337 views

First countable, second countable, and Lindelöf

I've studying the countability chapter in Schaum's General Topology and in the book, the author goes over first countable, second countable, and Lindelöf spaces. Once the author covers the definitions ...
SunRoad2's user avatar
  • 661
1 vote
1 answer
797 views

Topological properties of Sorgenfrey line.

Consider $\mathbb R$ with lower limit topology (generated by taking $[a,b)$ intervals as basis).The topological space thus generated is called Sorgenfrey line.What are some interesting properties of ...
Kishalay Sarkar's user avatar
3 votes
1 answer
454 views

Verifying proof that in a first countable space if $x_n\to x$ and $f(x_n)\to f(x)$, then $f$ is continuous

Suppose $X$ is a first-countable space and assume $f:X\to Y$ has the property that for every converging sequence in $X$: $x_n\to x$ the corresponding sequence $f(x_n)$ converges to $f(x)\in Y$, show ...
yotam maoz's user avatar
0 votes
0 answers
47 views

Is there a topology on natural numbers being not first countable? [duplicate]

A topology on a countable set may be uncountable, such as discrete topology on natural numbers. Although that topology has uncountably many set, it is first countable, indeed, second countable. I ...
Dr. Ufuk Kaya's user avatar
4 votes
1 answer
145 views

First countability of $[0,1]^\mathbb{R}$

The proofs I know for the fact that the space $[0,1]^\mathbb{R}$ is not first countable, use the product topology in some step of the demonstration. (Reference) So, I would like to know if this space ...
Mrcrg's user avatar
  • 2,777
0 votes
1 answer
234 views

First countable compact spaces are sequentially compact, how weaker assumption ease the proof? [closed]

A first countable, countably compact space is sequentially compact José Carlos Santos's answer Let $(x_n)_{n\in\mathbb N}$ be a sequence of elements of $X$. There are two possibilities: There is ...
Jale'de jaled's user avatar
0 votes
0 answers
90 views

Countable set if the cardinal number is smaller than aleph null

I have a question about countability od a set. I am confused with this problem, it seems like it holds but i am not sure. If a set $A$ has cardinal number smaller than the cardinal number of the set ...
MATH14's user avatar
  • 347
1 vote
0 answers
118 views

When is an absolute Galois group first-countable?

Let $K$ be a field, $K^{sep}$ its separable closure, and $G = Gal(K^{sep}/K)$ the absolute Galois group of $K$. What are some algebraic conditions on $K$ that ensure that $G$ is first-countable? This ...
frafour's user avatar
  • 3,025
0 votes
1 answer
72 views

Interior of a set and first countability

Summary: Let $(X,\tau)$ be a topological space and $A\subseteq X$. Is the first countability of $X$ needed for proving that any point $x$ is in the interior of $A$ iff every sequence in $X$ converging ...
Hermis14's user avatar
  • 2,597