Continuous bijections vs. Homeomorphisms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:05:28Z http://mathoverflow.net/feeds/question/87834 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87834/continuous-bijections-vs-homeomorphisms Continuous bijections vs. Homeomorphisms Ramiro de la Vega 2012-02-07T19:35:47Z 2012-02-07T19:35:47Z <p>This is motivated by an <a href="http://mathoverflow.net/questions/30661/non-homeomorphic-spaces-that-have-continuous-bijections-between-them" rel="nofollow">old question</a> of Henno Brandsma. </p> <p>Two topological spaces $X$ and $Y$ are said to be <em>bijectively related</em>, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. Let´s denote by $br(X)$ the number of homeomorphism types in the class of all those $Y$ bijectively related to $X$. </p> <p>For example $br(\mathbb{R}^n)=1$ and also $br(X)=1$ for any compact $X$. Henno´s question was about nice examples where $br(X)>1$. The list wasn´t too long, but all the examples in there also satisfied $br(X) \geq \aleph_0$. So here is my question:</p> <blockquote> <p>Is there a topological space $X$ for which $br(X)$ is finite and bigger than $1$?</p> </blockquote>