# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...