query: Why can't the Klein bottle embed in $\mathbb{R}^3$?
positive_passages.docid: 147310
positive_passages.text: How does one prove that the Klein bottle cannot be embedded in $R^3$?
negative_passages.docid: 148569
negative_passages.text: Embeddings $A → B → A$, but $A \not\cong B$?
negative_passages.docid: 133085
negative_passages.text: Is $\mathbb{R}^\omega$ endowed with the box topology completely normal (or hereditarily normal)?
negative_passages.docid: 93669
negative_passages.text: Is every embedded submanifold globally a level set?
negative_passages.docid: 88051
negative_passages.text: $\mathbb R = X^2$ as a Cartesian product
negative_passages.docid: 84189
negative_passages.text: Is $\mathbb{C}$ equal to $\mathbb{R}^2$?
negative_passages.docid: 67244
negative_passages.text: Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?
negative_passages.docid: 95876
negative_passages.text: Number of vertices in the Klein bottle triangulation given in "Topology and geometry for physicists" by Nash and Sen
negative_passages.docid: 67805
negative_passages.text: Why is fundamental group insufficient to classify manifolds?
negative_passages.docid: 85332
negative_passages.text: Is $\mathbb{R}^4$ minus a line simply connected?
negative_passages.docid: 51268
negative_passages.text: Why can't a Meredith Graph be Hamiltonian?
negative_passages.docid: 19499
negative_passages.text: Why is cross product only defined in 3 and 7 dimensions?
negative_passages.docid: 88723
negative_passages.text: Why does Van Kampen Theorem fail for the infinite earring space?
negative_passages.docid: 78085
negative_passages.text: A non orientable closed surface cannot be embedded into $\mathbb{R}^3$
negative_passages.docid: 74330
negative_passages.text: $\mathbb{R}^n$ not homeomorphic to open subset of $\mathbb{R}^m$
negative_passages.docid: 42883
negative_passages.text: Cross product in $\mathbb R^n$
negative_passages.docid: 51480
negative_passages.text: How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?