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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.

A topological space $X$ is called first countable if every point has a countable neighborhood base. Equivalent formulation using cardinal functions: $X$ has countable character, i.e., $\chi(X)\le\aleph_0$.

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Is the cofinite topology on an uncountable set first countable?

Let $X$ be any uncountable set with the cofinite topology. Is this space first countable? I don't think so because it seems that there must be an uncountable number of neighborhoods for each $ x \in X$. But I am not sure if this is true.
Jmaff
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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 $(x_n,y_n)_n$ is a sequence that is "limit-dense" in $X\times…
Chris Culter
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Why is $\mathbb{R}/{\sim}$ not first countable at $[0]$, where $x \sim y \Leftrightarrow x = y\text{ or }x,y \in \mathbb{Z}$?

I was searching for an easy example of a quotient space $X/{\sim}$ which is not second countable even though $X$ is second countable. I have found an example in the following answer, but I do not quite understand it: When is the quotient space of a…
asd
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Bases having countable subfamilies which are bases in second countable space

I don't understand a proof right at the beginning of this document found here. This proof is on why any base for the open sets in a second countable space has a countable subfamily that is a base. My trouble comes in the second paragraph of the…
fourtops
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Second Countable, First Countable, and Separable Spaces

Upon further studying Topological Spaces, I understand: If a space $X$ has a countable dense subset, then $X$ is a separable space. A space $X$ is first countable provided that there is a countable local basis at each point of $X$. A space $X$ is…
Kevin_H
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Partial limits in general topological spaces

Let $X$ be a general topological space, let $\{x_n\}_{n=1}^\infty\subseteq X$ be a sequence, and let $y\in X$. Suppose that for every $V\subseteq X$ open neighborhood of $y$, the set $\{n\in\mathbb{N}\mid x_n\in V\}$ is infinite. Does it necessarily…
User271828
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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...
RyArazi
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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 interval. Is $X$ first countable? I have read…
Qomo
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Mrówka spaces are first-countable

The construction of a Mrówka space (a $\Psi$-space) is not clear for me. Because of this I could not see why it is first-countable, locally compact, and Hausdorff. Coud you give me some help about this space?
user89330
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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 Prokhorov metric. However, this (a priori) doesn't…
Wooyoung Chin
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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 that a first countable Hausdorff space is compactly…
Barbara
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First countable + separable imply second countable?

In topological space, does first countable+ separable imply second countable? If not, any counterexample?
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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, there exists a neighborhood $U'$ of $p$ such…
user643073
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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 $\mathbb{N}$ has the discrete topology and $\{0,1\}$ has…
Mapply
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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 $x\in \mathbb{R}$, there exists a countable neighborhood…
sleeve chen
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