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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
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Does there exist a topological space $X$ such that $X^2$ and $[0,1]$ are homeomorphic?
A product $X\times Y$ of two connected spaces $X$ and $Y$ each with at least two points cannot have a cut point. Let $(a,b)$ be any point in $X\times Y$ and we'll show that $Z = (X\times Y)-\{(a,b)\}$ …