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 $f:X\rightarrow Y$ 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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106 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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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
67 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
34 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
28 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
36 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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26 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
49 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
33 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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40 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
52 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 ...
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2answers
35 views

Baby Rudin 2.12 vs power set of N [closed]

Baby Rudin 2.12 states that union of countable number of countable sets is countable. Doesn’t this contradict that the power set of $N$ is uncountable? Can somebody please explain? What I mean is: ...
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1answer
35 views

Can $X \setminus Y$ be first category? For $Y$ a proper linear subspace of a completely metrizable TVS $X$

So I got $0$ credit on one of my school problems. The comment was that if $Y$ is infinite dimensional, I cannot assume it has a countable basis. Hence I have absolutely no ideas how to solve this ...
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1answer
67 views

Locally euclidean and first countability

Suppose $X$ is a topological space that is locally euclidean of dimension some $n \in \Bbb{N}$. Show that $X$ is first countable. My attempt: Let $p\in X$ and $U$ a neighborhood of $p$. By assumption,...
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31 views

Continuous open maps and first countability

Problem: Let $f:X\rightarrow Y$ be continuous and open. Suppose $X$ satisfies first countability axiom. Show that $f(X)$ satisfies first countability axiom. My attempt: Let $b\in f(X)$ So there is an ...
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67 views

First countable space

Theorem: A subspace of a first countable space is first countable. My proof: Let $X$ be a first countable space. So for each p $\in X$, there exists a countable neighborhood basis for $X$ at $p$. Let ...
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3answers
69 views

Prove that the set of all differentiable functions 𝑓:[0,1]β†’ [0,1] is uncountable.

Prove that the set of all differentiable functions 𝑓:[0,1]β†’ [0,1] is uncountable. In my notes, I have something like: Consider $x \in {[0,1]}$, $f_{x} (t) = x$ for all $t$ . $\{h_{x} | x ∈ [0,1] \}=...
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1answer
112 views

Showing that a first countable space is Hausdorff if and only if every sequence converging in $X$ has a unique limit

Show that a first countable space $X$ is Hausdorff if and only if every sequence converging in $X$ has a unique limit. My Proof $X$ is Hausdorff, then it is trivial that every convergent sequence has ...
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89 views

Finite subset of real line is compact

Prove or disprove the following statements: A) A finite subset of the real line is compact. B) A countable subset of the real line is compact. For part A) I said, A finite subset of real line has ...
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66 views

Proof that the quotient space of $\mathbb{R}^2/L$ where $L$ is a line passing through the origin is not first countable

Let $L$ be a line in the plane $\Bbb R^2$ passing through origin. How would you prove that the quotient space of $\frac{\mathbb{\Bbb R}^2}{\sim}$ where $\sim$ is equivalence relation defined by $a\sim ...
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39 views

Is $\mathbb{Q}^n$ countable?

I was proving that $\mathbb{R}^n$ is separable and I found out that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ but I could not figure out the proof of $\mathbb{Q}^n$ countability.
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127 views

proof for sequentially continuous function with a first countable domain is continuous

Assuming the axiom of countable choice any function $f$ from a first countable space $X$ to $Y$ that is sequentially continuous is necessarily continuous. The gist of the proof I was thinking is that ...
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1answer
42 views

Rudin 4.17. Discontinuous points on a real function can only be at most countable

Could someone please explain to me the logic of the proof of the following problem? Exercise 4.17 Let $f$ be a real function defined on $(a,b)$. Prove that the set of points at which $f$ has a simple ...
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75 views

Example of topological space that is not first countable and also is not separable

I want to find a topological space that is not first countable and is not separable. Cofinite topology is not first countable but is separable. Please help me.
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1answer
163 views

Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Here is Theorem 30.1 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a topological space. (a) Let $A$ be a subset of $X$. If there is a sequence of points of $A$ converging to $x$, ...
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35 views

Limit points of sets in a first-countable space have a sequence converging to them

I have written a rudimentary proof of the title, but I'm not sure just how correct -or incorrect- it is. I'm fairly new to topology, and frankly I always feel out of my element when it comes to ...
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67 views

Theorem : Every countably compact subset of a first countable Hausdorff space is closed. What happens if the first countability condition is dropped?

Definition : A topological space is said to be countably compact if every countable open cover of the space admits a finite subcover. The Proof of the Theorem stated in the title uses the first ...
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1answer
29 views

Continuity of countable projection from non-first countable topological space

This might be trivial, but I just want to make sure I got this right: Let $X$ be a metric space and $I$ an uncountable index set. Let us consider $X^I$ with the product topology (of course, the ...
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1answer
35 views

Does a sequentially continuous function take its supremum on compacts?

Consider the following situation: Let $X$ be a separable metric space [if this helps: I am mainly interested in the case $X = \mathcal{P}(\mathbb{R}^d)$, the space of all Borel-probability measures on ...
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1answer
36 views

Countable related question

If $S$ is a countable subset of $R^2$, show that for any two points $x, y \in R^2 \setminus S$, there is a parallelogram in $R^2\setminus S$ having $x, y$ as opposite vertices. What can I do for this ...
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1answer
22 views

Nested interestection forms neighbourhood basis

Let $X$ be a topological space and $x \in X$. Suppose that there exists a countable collection $(U_n)_{n \geq 1}$ of open sets such that $U_{n + 1} \subseteq U_n$ and $\bigcap\limits_{n \geq 1} U_n = \...
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1answer
97 views

Does first countable imply equivalence of sequential and limit point compactness?

Steen and Seebach say that: If a topological space $X$ is first countable, sequentially compactness is equivalent to limit point compactness in $X$. Take $X=\mathbb{N}\times\{0,1\}$, where $\mathbb{...
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Given set U is first countable or not?

In $\mathbb{R}$ with usual topology ,the set $U =\{ x \in \mathbb{R} : -1\le x \le 1 , ,x \neq 0\}$ is Choose the correct statement $a)$ Neither hausdorff nor First counatble $b)$ Hausdorff $...
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1answer
144 views

What is the proof that first countable is sufficient to say that sequentially closed implies closed

I've seen this statement quoted many times but I've been unable to find a proof of the statement. I've been attempting to prove it myself with the following method: Let $X$ be a topological space, ...
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1answer
199 views

Neighborhood basis in the discrete metric space and first countability

Consider the topological space $\mathbb{R}$ with discrete metric $$d(x,y) = \begin{cases} 1 & x\neq y\\ 0 & x=y \end{cases}$$ We know that the metric space is first countable. So for each ...
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1answer
92 views

A first countable hemicompact space is locally compact

Prove: A first countable hemicompact space is locally compact. A topological space $(X,\tau)$ is said to be hemicompact if it has a sequence of compact subsets $K_n$, $n \in \mathbb{N}$, such that ...
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186 views

Is the topology of weak convergence of probability measures first-countable?

Let $S$ be a separable metric space, and $\mu,\mu_1,\mu_2,\ldots$ be Borel probability measures on $S$. We know that $\mu_n \to \mu$ weakly if and only if $\pi(\mu_n,\mu) \to 0$ where $\pi$ is the ...
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1answer
248 views

The continuous image of a First Countable Space need not be First Countable (Willard 16.B.1)

In Stephen Willard's General Topology appears the following exercise: A quotient of a second countable space need not be second countable (for each $n\in \mathbb{N}$, let $I_n$ be a copy of $[0,1]$ ...
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586 views

A first countable, countably compact space is sequentially compact

Let $(X, \mathcal{T})$ be a space that is both first countable and countably compact (every countable open cover has a finite cover). Show that $X$ is sequentially compact (every sequence has a ...
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1answer
62 views

If a space is first countable, does that imply it is locally compact? [closed]

If $X$ is a first countable space can I then somehow show that $X$ is also locally compact? Or are there counterexamples? Thanks for your time and best regards.
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1answer
18 views

Let $(X,\tau_x)$ first counable space , $(Y,\tau_y)$, prove that pra image f of an open set in Y, contains open set

Let $(X_1, \tau_1)$ be a first countable space, $(X_2,\tau_2)$ topological space, function $f\ :\ (X_1, \tau_1) \rightarrow (X_2,\tau_2)$ , V be any open set in $X_2$ prove that $f^{-1}(V)$ contains ...
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0answers
30 views

Check properties of the topology $\{U \subseteq \mathbb{R} \mid \forall x \in U: \exists \epsilon > 0: [x, x + \epsilon[ \subseteq U\}$

Consider the topological space $(\mathbb{R}, \mathcal{T}:=\{U \subseteq \mathbb{R} \mid \forall x \in U: \exists \epsilon > 0: [x, x + \epsilon[ \subseteq U\})$. Is this space separable? Is it ...
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1answer
305 views

If X satisfies the first or second countability axiom then $F(X)$ satisfies the same condition.

I have 2 questions related to the proof given here, In I, why to propose $\{F(B_i) \cap F(X)\}_{i \in I}$ as a basis for $f(p)$ when $\{B_i\}_{i \in I}$ is a basis for $p$? That is what we want to ...
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430 views

Prove that $\prod X_\alpha$ is first (second)-countable if and only if $X_\alpha$ is first (second)-countable, $\forall \alpha\in I.$

Let $I\neq\emptyset$ numerable and $(X_\alpha,\tau_\alpha)$ a family of topological spaces. Prove the following. $\displaystyle\prod X_\alpha$ is first-countable if and only if $X_\alpha$ is first-...
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1answer
68 views

First-order topologies, limit-points and convergent sequences

Let $(X,T)$ be a topological space, $D\subseteq X$ be a subset, and $x\in X$ have a countable neighbourhood basis $(U_k)$. I want to know how I can prove the following statement: If x is a limit-...
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1answer
44 views

Each $C^*$-embedded subset $S$ of a first countable Tychonoff space $X$ is closed.

Proposition. Each $C^*$-embedded subset $S$ of a first countable Tychonoff space $X$ is closed. Proof. Let $S$ be a non-closed subset of the space $X$. Pick a point $x_0\in \overline{S}\setminus S$ ...
4
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1answer
86 views

How many real function are there?

There are more numbers in $\mathbb{R}$ than in $\mathbb{N}$. There are as many vectors in $\mathbb{R}^n, n \in \mathbb{N}$ as numbers in $\mathbb{R}$. How many real functions are there? If I denote $\...
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1answer
49 views

Infinite Sets (1st yr Uni) and Axioms

This is my first question, so please be easy on me if it's too "conversational". I've been asked to consider how the axioms of ZFC Set Theory, and any that underlie it, could influence the way in ...
3
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1answer
360 views

Prob. 4, Sec. 21, in Munkres' TOPOLOGY, 2nd ed: First countability of $\mathbb{R}_l$ and of the ordered square

Here is Prob. 4, Sec. 21, in the book Topology by James R. Munkres, 2nd edition: Show that $\mathbb{R}_l$ and the ordered square satisfy the first countability axiom. (This result does not, of ...