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If a topological space X is not first countable, is it second countable?

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#general-topology #second-countable #first-countable
134
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Definition of compactness in terms of convergent sequences.

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#sequences-and-series #general-topology #compactness #first-countable
10.2k
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Second Countable, First Countable, and Separable Spaces

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#general-topology #separable-spaces #second-countable #first-countable
191
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Are the names such as $N_1$-space, $N_2$-space used for various countability axioms?

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#general-topology #reference-request #terminology #second-countable #first-countable
1.7k
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Why is $\mathbb{R}/{\sim}$ not first countable at $[0]$, where $x \sim y \Leftrightarrow x = y\text{ or }x,y \in \mathbb{Z}$?

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#general-topology #quotient-spaces #first-countable
2.9k
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First countable + separable imply second countable?

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#general-topology #examples-counterexamples #separable-spaces #second-countable #first-countable
1.7k
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Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

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#general-topology #examples-counterexamples #first-countable
803
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A non metric first countable topological space

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#general-topology #metric-spaces #examples-counterexamples #first-countable
1k
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Properties of Infinite set on co-finite topology and Countable set on co-countable topology

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#general-topology #proof-verification #second-countable #separable-spaces #first-countable
137
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A neighborhood basis in an arbitrary topological space for which the inclusion is a total order

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#general-topology #first-countable
955
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Mrówka spaces are first-countable

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#general-topology #compactness #separation-axioms #first-countable
1.1k
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T4 and first countable topology that is non metrizable

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#general-topology #examples-counterexamples #first-countable
10.6k
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Is the cofinite topology on an uncountable set first countable?

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#general-topology #first-countable
339
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countability of a set of uncountable many real intervals?

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#real-analysis #set-theory #real-numbers #first-countable
408
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basis of a second countable, topology

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#general-topology #second-countable #first-countable

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