# Questions tagged [first-countable]

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

112 questions
Filter by
Sorted by
Tagged with
0answers
44 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 ...
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 ...
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 ...
0answers
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 ...
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 ...
0answers
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?
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 ...
0answers
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 ...
0answers
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 ...
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 ...
1answer
34 views

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 ...
3answers
54 views

1answer
27 views

1answer
64 views

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 ...
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 $f\circ\nu$ ...
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 ...
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 ...
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. ...
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 ...
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 ...
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.
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 ...
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 ...
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?
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 ...