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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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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1answer
39 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 ...
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1answer
36 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 ...
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24 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 ...
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1answer
62 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 ...
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7 views

Proof Check- Cartesian Product of countable sets.

I feel like my proof has holes somewhere. Like is there any explanation needed for why Ap x Ak+1 is countable?
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1answer
29 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 ...
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20 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 ...
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38 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 ...
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1answer
20 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 ...
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1answer
34 views

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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1answer
69 views

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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4answers
233 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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1answer
42 views

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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3answers
54 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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1answer
34 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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1answer
42 views

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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1answer
52 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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1answer
27 views

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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2answers
28 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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0answers
37 views

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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1answer
64 views

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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1answer
39 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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1answer
49 views

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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1answer
32 views

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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1answer
50 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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1answer
45 views

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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1answer
135 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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2answers
68 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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1answer
49 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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1answer
38 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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1answer
70 views

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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2answers
122 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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0answers
39 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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1answer
43 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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1answer
66 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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1answer
45 views

Problem with extension of a continous function

Let $X$ a first-countable Topological Space, let $Y$ an Hausdorff Topological Space, let $A\subset X$ a subset ot $X$ and let $f:A\rightarrow Y$ a continous function. Prove that, if there is an ...
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1answer
91 views

First countable property for $\operatorname{spec}(\mathbb{C}[x])$ with Zariski topology

I'm interested in this particular property which i've developed a proof and i would like to know if my thoughts are correct. I've only considered the subspace $X = \operatorname{spec}(\mathbb{C}[x])\...
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1answer
26 views

If $f:X\rightarrow Y$ is not a constant function and if $X$ is first countable then $f$ is not continuous in any isolated point of $X$.

Conjecture If $X$ is firt countable and if $f:X\rightarrow Y$ is a function then $f$ is not continuous at $x_0$ if this is an isolated point for $X$. If $X$ is first countable then there exist a ...
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2answers
137 views

If $X$ and $Y$ are first countable then $\lim_{x\to x_0}f(x)=y_0$ iff $\lim_{n\to+\infty}f(x_n)=y_0$ for any $x_n\to x_0$

Definition The limit of a function $f:X\to Y$ as $x$ approaches at the limit point $x_0$ is $y_0$ if and only if any net $\nu:\Lambda\to X\setminus\{x_0\}$ converging to $x_0$ is such that $f\circ\nu$ ...
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1answer
111 views

Are the set of all convergent geometric series whose sum is a rational number is countable? [closed]

I tried this way: As the sum of convergent geometric series is $\frac{a}{1-r}$ and $-1<r<1$. Moreover sum is also a rational number. So $a$ and $r$ should be rational numbers. As rational ...
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1answer
69 views

Proving $B(A)$ is countable.

Can I get feedback/help with my proof please? Thanks! Let $A$ be a subset of $\mathbb{R}^n$ for $n\ge 1.$ Let $B(A)$ denote the points of $A$ such that $p\in B(A)$, then there is an open set $U$ with ...
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1answer
36 views

Showing that no point of $X^*$ is a point at which $\beta X$ is first countable.

I am having a tough time with this problem. It is from Munkree's Topology book. I am unsure where my proof is heading. Can someone please help me prove the problem? Thank you for your time and help. ...
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1answer
46 views

Prob. 12, Sec. 30, in Munkres' TOPOLOGY, 2nd edition: Every continuous open image of a first / second countable space is also first / second countable

Here is Prob. 12, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Let $f \colon X \rightarrow Y$ be a continuous open map. Show that if $X$ satisfies the first or the second ...
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1answer
51 views

Prob. 1 (b), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Example of a not-first-countable space with every singleton set being a $G_\delta$ set

Here is Prob. 1, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: (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 ...
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1answer
28 views

In which of the three topologies does $X$ have a countable basis?

Question: In which of the three topologies does $X$ have a countable basis? Below is the way I did....Can someone verify my proof ? Let me know if there is any concern or questions.
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1answer
128 views

Radially open set topology is separable?

Consider the topology of radially open sets on $\mathbb{R^2}$ : a set $S$ is radially open iff, for every $x\in S$ and every line $L\subset\mathbb{R^2}$ that contains $x$, $S$ contains an open segment ...
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1answer
68 views

Conclude that the following topological vector space is not first-countable

Suppose that we have a normed vector space $(X,\|\cdot\|)$. We endow $X$ with a (locally convex) topology $\tau$ such that any $\tau$-convergent is norm bounded. Suppose that there exists a countably ...
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1answer
41 views

About countable set

Let $f$ be a function in [a,b] with a countable break points (note by $S$). Is it true that the closure of S is countable?
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1answer
72 views

I want to prove that $\mathbb{Q}^{\infty}$ is not first countable. [closed]

I don't understand the hint which is similar to the case in $\mathbb{R}^{\omega}$ in box topology. Is there a subset of $\mathbb{Q}^{\infty}$ which does not satisfy the converse of sequence lemma? Or ...