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Countable basis but uncountably many connected components

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#general-topology #examples-counterexamples #connectedness #second-countable #locally-connected
827
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Rudin: A compact metric space $K$ has a countable base, therefore $K$ is separable.

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#general-topology #proof-verification #metric-spaces #compactness #second-countable
598
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If a topological space $S$ is second-countable, must necessarily every quotient space of $S$ be second-countable?

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#general-topology #examples-counterexamples #quotient-spaces #second-countable
4.5k
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Separable implies second countable

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#general-topology #separable-spaces #second-countable
1k
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Show that the Sorgenfrey line does not have a countable basis.

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#general-topology #second-countable #sorgenfrey-line
133
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Generalization of G-delta Sets

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#general-topology #second-countable
164
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Secound Countable spaces and perfect maps

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#general-topology #second-countable
2k
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Lindelöf and second countable spaces

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#real-analysis #general-topology #second-countable #lindelof-spaces
299
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Proof verification: Every separable metrizable space is second-countable.

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#general-topology #proof-verification #separable-spaces #second-countable
34
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Countable base for the one point compactification topology on $\mathbb{C}$

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#general-topology #second-countable
100
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Preservation of Second Countability

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#general-topology #solution-verification #geometric-topology #second-countable
37
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Disjoint spaces and Countability

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#general-topology #solution-verification #second-countable
85
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Is every compact subset of a second countable space closed?

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#general-topology #compactness #second-countable
566
Views

How can I prove that compact subset iff sequentially compact in second countable topological spaces?

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#sequences-and-series #general-topology #metric-spaces #compactness #second-countable
750
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Second countable and sequentially compact topological spaces are compact. Proof

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#sequences-and-series #general-topology #compactness #second-countable

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