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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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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 ...
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22 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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1answer
27 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
18 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
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 ...
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1answer
33 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
17 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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42 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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39 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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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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57 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 ...
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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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174 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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225 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
41 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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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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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
191 views

Continuous function and first and second countable spaces

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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2answers
212 views

Product topology and countable topology spaces

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 ...
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1answer
51 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
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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 $x_0\in \overline{S}\setminus S$ ...
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1answer
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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
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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 ...
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1answer
165 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 ...
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1answer
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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1answer
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 $B$ ...
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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?
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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,...
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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 $R_i$ ...
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1answer
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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. ...
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2answers
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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 ...
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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. ...
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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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1answer
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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 ...
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2answers
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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 ...
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Constraining a dense sequence on a product space, one factor at a time

Slogan: Given a sequence on $X\times Y$, can we choose subsequences to fix the limit in $X$ while leaving the behavior on $Y$ free? Details: Suppose $X$ and $Y$ are topological spaces and $(x_n,y_n)...
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1answer
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The family of non-decreasing functions from the segment info itself endowed with the topology of pointwise convergence is first countable

Using the fact that the only type of discontinuity compatible with monotonic function is the jump discontinuity, I showed that $A:=\{a\in[0,1]:f\textrm{ is discontinuous at }a\}$ is enumerable for ...
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A neighborhood basis in an arbitrary topological space for which the inclusion is a total order

Given a first countable space it is always possible to build a neighborhood basis of a point for which the inclusion is a total order. Let's say $\lbrace V_n \rbrace _{n \in \mathbb{N}}$ is any ...
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1answer
308 views

Properties of Infinite set on co-finite topology and Countable set on co-countable topology

I am trying to verify some of the properties of infinite set on co-finite topology and countable set on co-countable topology but it is proven to be very tricky because I cannot visualize the spaces. ...
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2answers
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A non metric first countable topological space [duplicate]

Every metric space is first countable, but what about the converse? Does it always hold? If not, can anyone give a counterexample? Thanks
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3answers
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Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it? Though there are ...
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2answers
1k views

First countable + separable imply second countable? [closed]

In topological space, does first countable+ separable imply second countable? If not, any counterexample?
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Are the names such as $N_1$-space, $N_2$-space used for various countability axioms?

In this question the OP mentioned "$N_i$-hierarchy for various countability axioms and also that the name $N_2$-space or $N_2$-property is used for second countable space. I did not encounter this ...
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1answer
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Topological counterexample: compact, Hausdorff, separable space which is not first-countable

I need an example for a compact, Hausdorff, separable space which is not first-countable. I tried to look for it for some time without success...
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1answer
272 views

Prove $\omega_1$ is first countable

Given a well order $(W,\le)$, where W is uncountable, and $\omega_1:= \{x\in W:$ only countably many $y \in X$ s.t. $y \le x\}$, prove $\omega_1$ is first countable. I saw a proof saying that $\{(a,x]...
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T4 and first countable topology that is non metrizable

Does anyone know any example of such topology?
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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 ...