query: Can someone explain why we take Second Countability and Hausdorffness conditions in Manifold definition?
positive_passages.docid: 72602
positive_passages.text: Why is important for a manifold to have countable basis?
negative_passages.docid: 108687
negative_passages.text: Open subsets in a manifold as submanifold of the same dimension?
negative_passages.docid: 47251
negative_passages.text: Non-compactness of $\mathbb{R}$ with the cocountable topology
negative_passages.docid: 67805
negative_passages.text: Why is fundamental group insufficient to classify manifolds?
negative_passages.docid: 17161
negative_passages.text: Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous?
negative_passages.docid: 94165
negative_passages.text: Why is a map to a smaller dimensional space not injective?
negative_passages.docid: 36184
negative_passages.text: Countability of disjoint intervals
negative_passages.docid: 10215
negative_passages.text: Differentiable manifolds as locally ringed spaces
negative_passages.docid: 143467
negative_passages.text: Continuous maps between compact manifolds are homotopic to smooth ones
negative_passages.docid: 17733
negative_passages.text: $X$ second countable locally compact Hausdorff implies $C(X)$ separable?
negative_passages.docid: 100476
negative_passages.text: Why does a Hausdorff but not countably compact space have an infinite closed discrete subset?
negative_passages.docid: 10513
negative_passages.text: What is the significance of the term "separable" in the context of countability properties?
negative_passages.docid: 48174
negative_passages.text: Are countable topological spaces second-countable?
negative_passages.docid: 150796
negative_passages.text: Subspace topology and order topology (2)
negative_passages.docid: 93389
negative_passages.text: Uncountable sets of Hausdorff dimension zero
negative_passages.docid: 19453
negative_passages.text: Invariance of domain for manifold
negative_passages.docid: 82989
negative_passages.text: T4 and first countable topology that is non metrizable