Is a topology determined by its convergent sequences? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:37:10Z http://mathoverflow.net/feeds/question/36379 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36379/is-a-topology-determined-by-its-convergent-sequences Is a topology determined by its convergent sequences? Tony 2010-08-22T15:59:47Z 2010-08-22T21:09:32Z <p>Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the same convergent sequences? </p> http://mathoverflow.net/questions/36379/is-a-topology-determined-by-its-convergent-sequences/36380#36380 Answer by Mariano Suárez-Alvarez for Is a topology determined by its convergent sequences? Mariano Suárez-Alvarez 2010-08-22T16:06:46Z 2010-08-22T16:06:46Z <p>The cocountable topology on an uncountable set is undistinguishable from the discrete topology if you can only use sequences.</p> http://mathoverflow.net/questions/36379/is-a-topology-determined-by-its-convergent-sequences/36382#36382 Answer by Pete L. Clark for Is a topology determined by its convergent sequences? Pete L. Clark 2010-08-22T16:19:19Z 2010-08-22T16:26:47Z <p>In a metric (or metrizable) space, the topology is entirely determined by convergence of sequences. This does not hold in an arbitrary topological space, and Mariano has given the canonical counterexample. This is the beginning of more penetrating theories of convergence given by <a href="http://en.wikipedia.org/wiki/Net_%28topology%29" rel="nofollow">nets</a> and/or <a href="http://en.wikipedia.org/wiki/Filter_%28mathematics%29" rel="nofollow">filters</a>. For information on this, see e.g. </p> <p><a href="http://math.uga.edu/~pete/convergence.pdf" rel="nofollow">http://math.uga.edu/~pete/convergence.pdf</a></p> <p>In particular, Section 2 is devoted to the topic of sequences in topological spaces and gives some information on when sequences are "topologically sufficient".</p> <p>In particular a topology <em>is</em> determined by specifying which nets converge to which points. This came up as a <a href="http://mathoverflow.net/questions/19285/how-do-you-axiomatize-topology-via-nets" rel="nofollow">previous MO question</a>. It is not covered in the notes above, but is well treated in Kelley's <em>General Topology</em>.</p> http://mathoverflow.net/questions/36379/is-a-topology-determined-by-its-convergent-sequences/36390#36390 Answer by Ady for Is a topology determined by its convergent sequences? Ady 2010-08-22T20:08:34Z 2010-08-22T20:08:34Z <p>Just another example. Consider the Banach space $\ell^{1}\left(\Gamma\right)$ , $\Gamma$ being an infinite set. Then the weak topology and the norm topology have the same convergent sequences (Schur' Theorem), while they are clearly distinct.</p> http://mathoverflow.net/questions/36379/is-a-topology-determined-by-its-convergent-sequences/36400#36400 Answer by David Spivak for Is a topology determined by its convergent sequences? David Spivak 2010-08-22T21:09:32Z 2010-08-22T21:09:32Z <p>There is a category of "sequential spaces" in which objects are spaces defined by their convergent sequences and morphisms are those maps which send convergent sequences to convergent sequences.</p> <p>As stated above, all metric spaces are sequential spaces, but so are all manifolds, all finite topological spaces, and all CW-complexes.</p> <p>To build this category, one actually just needs to look at the category of right $M$-sets for a certain monoid $M$. Consider first the "convergent sequence space" $S:=${$\frac{1}{n}|n\in{\mathbb N}\cup${$\infty$}}$\subset {\mathbb R}$. In other words $S$ is a countable set of points converging to 0, and including $0$. Let $M$ be the monoid of continuous maps $S\to S$ with composition. Then an $M$-set is a "set of convergent sequences" closed under taking subsequences. </p> <p>The category of $M$-sets is a topos, so it has limits, colimits, function spaces, etc. And every $M$-set has a topological realization which is a sequential space.</p>