# Questions tagged [first-countable]

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

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

Here is Theorem 30.2 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 ...
1answer
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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 ...
1answer
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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 ...
1answer
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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 ...
1answer
27 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 ...
1answer
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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 ...
1answer
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3answers
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1answer
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### 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 ...
1answer
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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 ...
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106 views

### Show that a sequence in $X$ converges $⇔$ it has a single cluster point

The problem says Let $X$ be a countably compact $T 2$ -space satisfying the first axiom of countability. Show that a sequence in $X$ converges $⇔$ it has a single cluster point. If x is the ...
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142 views

### Prove that a countably compact, first countable $T_{2}$ space is regular.

The problem says Prove that a countably compact, first countable $T_{2}$ space is regular. $T_{2}$ space = space Hausdorff I have I take an open $U$ such that $x \in U$ and $B \subset U$ with $B$ ...
1answer
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### Is a set of circles in $\mathbb{R}^2$ such that no two circles (not discs) overlap necessarily countable or possibly uncountable?

Is a set of circles in $\mathbb{R}^2$ such that no two circles (not discs) overlap necessarily countable or possibly uncountable?
1answer
197 views

### basis of a second countable, topology [closed]

Prove that every basis of a second countable space contains a countable subfamily which is also a basis. *I try with the intersection of the bases, the open ones of the finite with those of the other,...
1answer
139 views

### countability of a set of uncountable many real intervals?

Consider a real line $R$ and for each $i ∈ R$ there is a real line, $R_i$, that intersects $R$ at $i$; the $R$ and $R_i$ share only the number $i$ and let none of $R_i$ intersect. Since each $R_i$ ...
1answer
30 views

### Existance of a countable base for $\mathbb R$

Prove that $\mathbb R$ seen as a $\mathbb Q$-vector space cannot have a countable basis. I think that it is obvious that the main idea to use is that $\mathbb Q$ is countable but $\mathbb R$ is not. ...
2answers
246 views

### To prove $\omega_1 \cup \{\omega_1\}$ is not first countable but $\omega_1$ is

The topological space $X=\omega_1\cup \{\omega_1\}$ is not first countable but $\omega_1$ is. Here $\omega_1$ denotes the set of all countable ordinals and $\{\omega_1\}$ is the first uncountable ...
1answer
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### Some question about first (or second) countable - first (second) “countabilization”

Given $X$ be a topological space (or maybe a topological group), I am curious about is there a coarser or finer topology of $X$ or some quotient space such that it become first (or second) countable. ...
1answer
288 views

### A first countable Hausdorff space is compactly generated

I know that even a non-Hausdorff first countable space is compactly generated, but I assume that adding the property that the space is also Hausdorff, there is an easier proof. How would you prove ...
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### Checking for countinuity and countability axioms for this topology on the real plane $\mathbb{R}^2$

In the real plane $\mathbb{R}^2$ is considered the topology $\tau$ whose basis consists in the open squares centered at $p$ with lengths $\varepsilon >0$, excluding the points on the two diagonals ...
2answers
104 views

### If every point of a compact Hausdorff space is the intersection of a nested sequence of open sets then is the space first-countable?

Let $X$ be a compact Hausdorff space such that for every $x \in X$ , there exist a nested sequence of open sets $\{U_n\}$ such that $\{x\}=\bigcap_{n=1}^\infty U_n$ , then is it true that $X$ is first ...
1answer
269 views

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### T4 and first countable topology that is non metrizable

Does anyone know any example of such topology?
2answers
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### Uncountable Cartesian product of closed interval

I have a question about product topology. Suppose $I=[0,1]$, i.e. a closed interval with usual topology. We can construct a product space $X=I^I$, i.e. uncountable Cartesian product of closed ...