# Questions tagged [first-countable]

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

159 questions
Filter by
Sorted by
Tagged with
48 views

### 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 ...
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 ...
• 183
1 vote
79 views

• 423
1 vote
69 views

### 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 $X$ ...
• 649
40 views

### 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 ...
• 483
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 ...
1 vote
33 views

### 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) ...
• 483
1 vote
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 ...
1 vote
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 $(x_{n})_{n \in \mathbb{N}}$ ...
• 623
1 vote
44 views

### 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-]...
• 6,078
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 ...
• 3,915
1 vote
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 ...
61 views

### 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 ...
• 1,800
1 vote
68 views

### 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. ...
• 95
1 vote
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 ...
• 95
87 views

### 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 ...
• 600
131 views

### 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 ...
• 2,325
1 vote
103 views

• 13k
1 vote
164 views

• 4,173
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 ...
• 4,797
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 ...
• 431
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 ...
• 1,981
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 ...
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 ...
• 906
1 vote
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 ...
• 241
48 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 ...
• 649
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 ...
• 337