Welcome to MathOverflow

Question

4

3

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?

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 $\{ \mathbb{R}^{n} \}_{n \geq 0}$ 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).

flag	asked Mar 8 2011 at 10:25 Sam Nolen 467●2●9

1

Your motivation question seems much more interesting! Related questions have been discussed on MO before, e.g. mathoverflow.net/questions/19152/… . The answers that most convinced me were the ones involving logic and computability. They suggest that the definition of a topology is useful because it is absurdly general, hence general enough to include nice things. But it is not necessarily geometrically natural. I think Grothendieck once expressed an opinion that the definition is "wrong" e.g. for homotopy theory? – Qiaochu Yuan Mar 8 2011 at 10:40

2

If you used quotients and topological sums, you would get sequential spaces. Using subspaces, quotiens and sums, you would get subsequential spaces. (S. P. Franklin, M. Rajagopalan: On subsequential spaces, Topology. and its Applications 35 (1990), 1–19) Your class will definiely be a subclass of the class of subsequential spaces. I am not sure about the precise characterization. – Martin Sleziak Mar 8 2011 at 11:25

This way you can get only separable spaces with finite dimension and I guess you can get all of them (?) – Anton Petrunin Mar 8 2011 at 15:48

2

@Anton Petrunin: Finite dimension can't be right, since every compact metrizable space is a quotient of the Cantor set, and that includes things like $[0,1]^{\aleph_0}$ . – Stephen S Mar 8 2011 at 21:17

What spaces can be obtained from $\mathbb{R}^{n}$ by taking quotient spaces and subspaces?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.