Topological Characterisation of the real line. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:59:07Z http://mathoverflow.net/feeds/question/76134 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76134/topological-characterisation-of-the-real-line Topological Characterisation of the real line. Suryateja 2011-09-22T16:27:30Z 2011-10-02T18:20:46Z <p>What is a purely topological characterisation of the real line( standard topology)?</p> http://mathoverflow.net/questions/76134/topological-characterisation-of-the-real-line/76139#76139 Answer by Gjergji Zaimi for Topological Characterisation of the real line. Gjergji Zaimi 2011-09-22T17:07:12Z 2011-09-26T11:21:28Z <p>Here are a few examples that came up in a first search. Ward in "The topological characterization of an open linear interval", Proc. London Math. Soc.(2) 41 (1936), 191-198 proved the characterization of the real line as a connected, locally connected separable metric space, such that every point is a strong cut point (removing it leaves precisely two connected components). Franklin and Krishnarao proved that in this characterization "metric space" can be relaxed to "regular space", "On the topological characterization of the real line", Department of Mathematics, Carnegie-Mellon University, Report #69-36, 1969.</p> <p>On a different note, Thron and Zimmerman prove in "A characterization of order topologies by means of minimal T0-topologies", Proc. Amer. Math. Soc. 27, (1971), 161-167, that order topologies $\tau$ on a set $X$ can be characterized as the topologies for which $(X,\tau)$ is $T_1$ and $\tau$ is the least upper bound of two minimal $T_0$ topologies. (Minimal here means that the open sets form a nested family of sets and that the complements of the point closures form a base for the topology.) Similarly the reals can be characterized as a connected, separable, $T_1$ space, and $\tau$ is the least upper bound of two noncompact minimal $T_0$ topologies.</p> http://mathoverflow.net/questions/76134/topological-characterisation-of-the-real-line/76140#76140 Answer by Valerio Capraro for Topological Characterisation of the real line. Valerio Capraro 2011-09-22T17:15:41Z 2011-09-22T17:15:41Z <p>Gjergji's answer looks very exhaustive. Let me just add the paper <a href="http://www.jstor.org/pss/2308632" rel="nofollow">http://www.jstor.org/pss/2308632</a> where they propose a topological characterization, also in this case using the order topology (the real line is the unique <em>linear</em> space which is separable, connected and having neither maximum nor minimum). Personally, I would avoid the order and I really like Ward's characterization.</p> http://mathoverflow.net/questions/76134/topological-characterisation-of-the-real-line/76145#76145 Answer by Theo Johnson-Freyd for Topological Characterisation of the real line. Theo Johnson-Freyd 2011-09-22T18:33:43Z 2011-10-02T18:20:46Z <p>Gjergi's answer is almost certainly the best possible. But I wanted to add another one, that's a bit more "modern" in its approach to characterization problems. Namely, consider the category whose objects consist of a topological space $X$ and a pair of (<b>edit:</b> distinct, closed) points $l_X,r_X \in X$. This category has a monoidal structure, given by $$(X,l_X,r_X) \otimes (Y,l_Y,r_Y) = \bigl( (X \sqcup Y) / (r_X = l_Y), l_X, r_Y \bigr).$$ Now consider the endofunctor $(-)^{\otimes 2}$ of this category that sends each object to its "tensor square". Recall that <i>coalgebra</i> for an endofunctor $F$ is an object $\mathcal X$ and a morphism $\mathcal X \to F (\mathcal X)$; given any endofunctor, there is a category of coalgebras. Unpacking when $\mathcal X = (X,l_X,r_X)$ and $F = (-)^{\otimes 2}$, a coalgebra is precisely a topological space $X$ with two marked points $l_X,r_X$, along with a continuous map $X \to X \sqcup_{r_X = l_X} X$ fixing the remaining marked points. </p> <p>The amazing result is that the category of coalgebras of this functor has a terminal object, and that terminal object is precisely the closed interval $[0,1]$, with $l = 0$ and $r = 1$. Then it's clear how to characterize $\mathbb R =$ the open interval: it is the final coalgebra for this functor, minus its two marked points.</p>