What spaces can be obtained from \$\mathbb{R}^{n}\$ by taking quotient spaces and subspaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:05:47Z http://mathoverflow.net/feeds/question/57805 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57805/what-spaces-can-be-obtained-from-mathbbrn-by-taking-quotient-spaces-and-s What spaces can be obtained from \$\mathbb{R}^{n}\$ by taking quotient spaces and subspaces? Sam Nolen 2011-03-08T10:25:05Z 2011-03-08T10:25:05Z <p>Is there a good characterization of the smallest collection of topological spaces which contains \$\mathbb{R}^{n}\$ for each \$n\$, and is closed under taking subspaces and quotient spaces?</p> <p>A bit of motivation: A friend of mine asked me to give an argument why the definition of a topological space is "right" or "natural", considered perhaps as a generalization of manifolds or cell complexes. While trying to answer him, I briefly wondered whether the collection of topological spaces is the closure of <code>\$\{ \mathbb{R}^{n} \}_{n \geq 0}\$</code> under certain operations, say taking subspaces and quotient spaces. I quickly realized that this is false in general, though (there are counterexamples which have very large cardinality or don't satisfy first or second countability). </p>