# Questions tagged [first-countable]

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

128 questions
Filter by
Sorted by
Tagged with
1 vote
30 views

• 1,379
21 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 ...
• 3,423
92 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 ...
• 329
27 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 ...
• 1,811
72 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 ...
38 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 ...
• 724
1 vote
52 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 ...
• 241
33 views

### 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 ...
• 445
28 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 ...
• 195
22 views

### 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 ...
• 974
93 views

• 1,497
88 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 $X$ ...
1 vote
463 views

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 ...
43 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. =========================...
• 603
66 views

• 603
1 vote
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 ...
• 347
1 vote
47 views

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.
1 vote
55 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.
• 171
1 vote
58 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 $\langle x_\delta\rangle_{\delta\in\Delta}$ ...
1 vote
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 ...
• 205
1 vote
85 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 ...
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.
1 vote
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 ...
• 365
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 ...
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 ...
• 3,341
1 vote
86 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.
• 753
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 ...
• 753
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 ...
• 753
1 vote
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 ...
• 753
### 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 ...