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Chain of compact sets in locally compact space

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#general-topology #proof-writing #compactness #second-countable
82
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Example of a topological space that has countably (but not finitely) many elements, is regular, but not 2nd countable

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#general-topology #second-countable
200
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$f:I\rightarrow X$ where $X$ is hausdorff show that $X$ is metrizable.

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#general-topology #metric-spaces #compactness #connectedness #second-countable
242
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Let $(E,d)$ be a metric space. Then $X$ is second-countable if and only if $X$ is Lindelöf if and only if $X$ is separable

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#general-topology #metric-spaces #solution-verification #second-countable
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Proof that Metric Spaces are Second Countable?

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#general-topology #metric-spaces #second-countable
78
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Topology counterexamples without ordinals

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#sequences-and-series #general-topology #compactness #second-countable #first-countable
75
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On second-countable space and its uncountable subset

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238
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Prob. 5 (a), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every metrizabe separable space is second-countable

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174
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Prob. 5 (b), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every metrizable Lindelof space is second-countable

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#general-topology #solution-verification #metric-spaces #second-countable #metrizability
95
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Theorem 30.3 of Munkres’ Topology

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#general-topology #solution-verification #proof-writing #alternative-proof #second-countable
88
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Definition of Countability in Munkres’ Topology

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#general-topology #definition #second-countable #first-countable
110
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Exercise 5, Section 30 of Munkres’ Topology

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#general-topology #solution-verification #proof-writing #second-countable #lindelof-spaces
276
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Exercise 4, Section 30 of Munkres’ Topology

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#general-topology #solution-verification #proof-writing #compactness #second-countable
232
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Concerning topological manifolds: Are paracompact and connected locally euclidian Hausdorff spaces always second-countable?

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#general-topology #manifolds #second-countable #paracompactness
335
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Why does satisfying the countability conditions make topological spaces so nice?

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#general-topology #soft-question #intuition #second-countable #first-countable

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