query: Is there a space $X$ such that $S^1$ is homeomorphic to $X \times X$?
positive_passages.docid: 20383
positive_passages.text: True or False: there is a space $X$ such that $S^1$ is homeomorphic to $X\times X$
negative_passages.docid: 80860
negative_passages.text: How to call a space in which every G$_\kappa$-set is open
negative_passages.docid: 22604
negative_passages.text: Can $R \times R$ be isomorphic to $R$ as rings?
negative_passages.docid: 74330
negative_passages.text: $\mathbb{R}^n$ not homeomorphic to open subset of $\mathbb{R}^m$
negative_passages.docid: 129081
negative_passages.text: Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
negative_passages.docid: 57816
negative_passages.text: $(S^1 \times [0,1])/\sim$ homeomorphic to unit disk $ D^2$
negative_passages.docid: 52556
negative_passages.text: Isomorphism of $S^1$.
negative_passages.docid: 107076
negative_passages.text: Prove $S^1$ is not homeomorphic to $S^2$ using connectedness
negative_passages.docid: 131082
negative_passages.text: Proof that $\mathbb R/[0,1]$ and $\mathbb R$ are homeomorphic
negative_passages.docid: 128301
negative_passages.text: Homeomorphism from square to unit circle
negative_passages.docid: 70082
negative_passages.text: Show: Quotient space is homeomorphic to unit sphere
negative_passages.docid: 80100
negative_passages.text: Boundary of product manifolds such as $S^2 \times \mathbb R$
negative_passages.docid: 2348
negative_passages.text: Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic
negative_passages.docid: 88714
negative_passages.text: Are there more general spaces than Euclidean spaces to have the Heine–Borel property?
negative_passages.docid: 126463
negative_passages.text: Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic?
negative_passages.docid: 27629
negative_passages.text: How to prove that $S^{2n+1}/S^1$ is homeomorphic to $\mathbb CP^n$ under a given identification
negative_passages.docid: 12883
negative_passages.text: Give an example of a topological space $X $ such that $S_3$ acts freely on $X $