Questions tagged [first-countable]
For questions about first countable topological spaces, i.e., space with countable local base at each point.
161
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For compact Hausdorff spaces, is countable pseudocharacter equivalent to first countable? [duplicate]
Let $X$ be a compact $T_2$ space. Is $X$ first countable if, and only if, $X$ has countable pseudocharacter?
Note: I have already proven that every $T_1$ first countable space has countable ...
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Prove that a product of first-countable topological spaces is first-countable iff only countably many of them don't have the trivial topology [closed]
I need to show that if $\{X_\alpha\}_{\alpha \in \Lambda}$ are all first-countable spaces, then $\prod_{\alpha \in \Lambda} X_\alpha$ is first-countable if and only if there exists a $\Lambda' \...
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countable set of subsets of $\mathbb{Z}^{d}$ [closed]
Is the set of finite subsets of $\mathbb{Z}^d$ which contain a prescribed vertex and are compact and connected, countable?
Hint: It is clear that it is not countable without the restriction that the ...
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Definition of compactness in terms of convergent sequences.
This is a question about first countable spaces. Topology of such spaces can be defined in terms of convergent sequences, and many topological properties of such spaces can be expressed in terms of ...
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If a topological space X is not first countable, is it second countable?
I know that topological space is first countable if each point has a countable neighborhood basis. A neighborhood basis at a point. Consider the following topological space, $X=R$ with $\mathcal T=\{A\...
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Are dual spaces to separable normed spaces first-countable?
In [1] I found the following theorem (roughly translated by me)
Satz 13.10 If $X$ is a separable normed $k$-vector space ($k = \mathbb R$ or $\mathbb C$) with continuous dual space $X'$, then the ...
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Proving first countability of $\mathbb{R}$ under the particular point topology.
I am trying to show that $\mathbb{R}$ is a first countable space with respect to the particular point topology defined by:
$$\mathcal{T} = \{I\subseteq \mathbb{R} : I = \emptyset \text{ or } p\in I\}$$...
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3
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Countably close topological spaces
Does every uncountable topological space has a different and mutually "countably close" space?
Define "countably close" space $S$ with respect to another space $S'$ both on the ...
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Topological spaces with countable open set difference
My question is regarding the existance of "countably close" open sets to those of a parotopological group (or any group equipped with a topology).
Consider a group $G$ equipped with a ...
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108
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Sequentially Compact implies Compact?
I know that in metric spaces sequentially compact is equivalent to compact. I also know that compact and sequentially compact for general topologies there is no relation between them. But I wanted to ...
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homeomorphism between limit point compact set and 1st countable hausdorff space
X is a limit point compact space and Y is a 1st countable hausdorff space. Then show that bijective, continuous map f:X->Y is a homeomorphosm
All we need to show is f is an open or closed map. I've ...
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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}\...
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Are sober noetherian spaces sequential?
A sequential topological space $X$ has a few different equivalent definitions:
$X$ is the quotient of a first-countable space
$X$ is the quotient of a metric space
Sequentially open subsets of $X$ ...
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Prove that the following properties are all finitely productive
The question goes as follows:
Prove that the following properties are all finitely productive
(1) $T_0$ and $T_1$
(2) Separable
(3) First Countable
(4) Second Countable
(5) Finite (i.e., the ...
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First-countable topological spaces
So I have a topology defined as follows:
$$ \tau = \{\mathbb{R} \} \cup \{ U \subset \mathbb{R} \ \ | \ \ 0 \notin U \}$$
I have already prooved that is a topology of $\mathbb{R}$ and that the local ...
