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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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\...
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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 ...
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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 $...
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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 ...
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$\prod_{n=1}^{\infty}{\mathbb{R}}$ 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 ...
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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 ...
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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 ...
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3 answers
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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 ...
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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 ...
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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 ...
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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 ...
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Two first countable $T_{3\frac{1}{2}}$ spaces having homeomorphic Stone-Čech compactifications are homeomorphic

It suffices to show that we can reconstruct $X$ from $\beta X$. Every $x \in X$ has $\chi(x, X)\le\aleph_0$. Since $\beta X$ is normal and has $X$ as a dense subspace, the character of every closed ...
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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 $\...
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1 answer
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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 ...
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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. ...
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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 ...
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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 ...
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2 votes
1 answer
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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 ...
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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 ...
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4 votes
1 answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1 answer
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Help needed in understanding an argument in Theorem 2.47 of topology by Wayne Patty

I am self studying topology from Wayne Patty and I have a question in Theorem 2.48 on page 90. Subsection is Weak topology and product topology. The question is that I am not able to understand how $\...
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1 answer
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Show that $Y$ is not first-countable at $[0]$

(a) Show that $X := \{0, 1\}^{\mathbb R}$ is not first-countable. (b) Let $X := \bigcup_{n=0}^∞ [2n, 2n + 1]$ be a union of countably many disjoint closed intervals. Let $Y$ be the quotient of $X$ ...
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1 vote
3 answers
463 views

Show that $X :=$ {$0, 1$}$^{\mathbb R}$ is not first-countable.

Show that $X :=$ {$0, 1$}$^{\mathbb R}$ is not first-countable. According to Munkres Topology: Definition: A space $X$ is said to have a countable basis at $x$ if there is a countable collection $\...
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1 vote
1 answer
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Let X be a topological space. a) If X is Hausdorff and it has an countable base, so X is normal?

Let X be a topological space. a) If X is Hausdorff and it has an countable base, so X is normal? b) If X is regular, and if each point of X has a fundamental countable neighborhood system and is ...
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2 votes
3 answers
79 views

Let $X$ be the disjoint union of copies of a subspace of $\mathbb{R}$. Is the quotient of $X$ by identifying accumulation points first countable?

Sorry the title isn't super clear, I ran out of characters. From Willard's General Topology, 16A.5: For each $n \in \mathbb{N}$, let $X_n$ be a copy of the subspace $\{0\} \cup \{\frac{1}{m}: m = 1, ...
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1 answer
41 views

Box topology first countability

I want to show that $\mathbb{R}^2$ with the topology $\tau_\mathcal{B}$ generated by the base $\mathcal{B} = \{U \times V | U \in \tau_1, V \in \tau_2\}$ where $\tau_1$ and $\tau_2$ are topologies for ...
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1 answer
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Let $X = [0,1]$ and $\sim$ have the following equivalence relation over $X$: is the quotient first countable?

Let $X = [0,1]$ and $\sim$ have the following equivalence relation over $X$: $$x\sim y\iff x = y\text{ or }x, y \in {]}0, 1{[}$$ Show that $X/{\sim}$ is not first countable. =========================...
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0 votes
1 answer
66 views

Let $X$ and $Y$ be topological spaces and $f: X \to Y$ a function....

Let $X$ and $Y$ be topological spaces and $f: X \to Y$ a function. a) Show that if $f$ is continuous, then for each $ x \in X$ and each sequence $ (x_n) _ {n \in \mathbb N}$ in $X$ such that $ x_n \...
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0 votes
1 answer
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Let I be a uncountable set and {$(X_i, \tau_{i} $): i $\in$ I} a family of topological spaces $T_1$ with at least two points

Let I be a uncountable set and {$(X_i, \tau_{i} $): i $\in$ I} a family of topological spaces $T_1$ with at least two points.Set f $ \in \prod_{i \in I} X_i $.Show that the subspace S = { g $\in$ $\...
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  • 603
1 vote
2 answers
44 views

Is this topology separable, first countable, second countable?

Consider a set $E$ with some point $p \in E$ and the topology: $T = \left \{ \emptyset \right \}\cup \left \{ C \subset E: p \in E \right \}$. Is this topology, separable, first countable or second ...
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1 vote
0 answers
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The Sequence Lemma and the Axiom of Countable Choice [duplicate]

Consider the following lemma (sometimes called Sequence Lemma) Let $X$ be a topological space, $A\subseteq X$ any subset and $x\in X$. If there is a sequence of points in $A$ converging to $x$, then $...
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1 vote
1 answer
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Some topological properties of "countable lines with one origin"

Let the countable lines with one origin to be a quotient space $CL = ([0, \infty) \times \mathbb N) / \sim$, where $[0, \infty) \times \mathbb N$ has a subspace topology of $\mathbb R^2$ and $0 \times ...
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-1 votes
1 answer
53 views

How can I prove that the cofinite topology on a countable set is first countable? [duplicate]

Is the cofinite topology on a countable set first countable? I know it is not first countable on R but I don't know how to prove that it is first countable on a countable set.
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1 vote
1 answer
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Show that there is a bijection between $2^{\mathbb{N}}$ and $\mathbb{R}$

I tried to do it with ternary/binary expansions by finding two injections and then use the Cantor-Bernstein-Schröder Theorem, but I wonder if there is some easier method to prove this.
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  • 171
1 vote
1 answer
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Closures and nets in topological spaces

Suppose $(X,\tau)$ is a topological space and $A\subset X$. Let $\overline{A}$ denote the closure of $A$ in $X$. Suppose $x\in A$. Then there exists a net $\langle x_\delta\rangle_{\delta\in\Delta}$ ...
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1 vote
1 answer
104 views

Zariski topology is not first countable on $\mathbb{R}$

Prove that the Zariski topology is not first countable on $\mathbb{R}$. All I'm able to show right now is that all the one-point sets $\left(\{a\}\subset \mathbb{R}^n\right)$ are closed as every ...
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1 vote
1 answer
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Countable Local Bases for a Metric Space

Let $(\Bbb{X}, \rho)$ be a Metric Space. Consider the Metric Topology $(\Bbb{X}, \tau_{\rho})$ and fix $x \in \Bbb{X}$. Why does $\mathscr{B}_x = \{B_{\rho}(x, \epsilon): \epsilon > 0\}$ form, in ...
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0 votes
1 answer
272 views

Is the cofinite topology on a countable set first countable?

Is the cofinite topology on a countable set first countable? I know it is not first countable on R but I don't know the case of a countable set.
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1 vote
2 answers
157 views

A property of a first-countable space

I am having trouble solving the exercise 4 in chapter 2, section 4 of Introduction to Topology, Gamelin and Greene, 2nd. Suppose a topological space $X$ satisfies the first axiom of countability, or ...
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1 answer
163 views

Prove on metrization of uncountable product [duplicate]

I am given the following problem: Show that an uncountable product of unit intervals is not first countable, and thus not metrizable. My answer would be that a), since the elements of the neighborhood ...
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3 votes
1 answer
65 views

Is every First Countable space the continuous image of a Pseudometric space under an open map?

In Engelking's General Topology, Exercise $4.2.\text{D}.(\text{a})$, we are asked to show that a $T_0$ space is $1^{st}$ Countable iff it is the continuous image of a metrizable space under an open ...
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1 vote
1 answer
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Is there a compact Hausdorff sequential space which is not first countable at a dense subset?

Let $X$ be a compact Hausdorff sequential space. Does $X$ have a dense subset of points which have a countable local base? Every example that I have seen so far has this property.
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3 votes
2 answers
177 views

Is a locally compact hereditarily Lindelof Hausdorff space first countable?

Is a locally compact hereditarily Lindelof Hausdorff space first countable? I was recently told that it is but I can't find any reference to what I would have thought would be a standard fact if it is ...
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2 votes
0 answers
51 views

Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space first countable on a dense set?

Let $Y$ be a locally compact, $\sigma$-compact, first countable Hausdorff space and $q:Y\to X$ a quotient map with $X$ Hausdorff. Suppose that $X$ is locally compact. Is $X$ first countable at a dense ...
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1 vote
1 answer
54 views

Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space always Frechet-Urysohn?

This question follows on from a previous one, which has been answered in the negative: Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space ...
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2 votes
1 answer
82 views

Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space always first countable?

Let $Y$ be a locally compact, $\sigma$-compact, first countable Hausdorff space and $q: Y\to X$ a quotient map with $X$ Hausdorff. Suppose that $X$ is locally compact. Is $X$ first countable? I have ...
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