Compact open topology on $\mathrm{Homeo}(X)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:49:53Z http://mathoverflow.net/feeds/question/58690 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58690/compact-open-topology-on-mathrmhomeox Compact open topology on $\mathrm{Homeo}(X)$ Olivier Bégassat 2011-03-16T22:09:19Z 2011-04-12T17:34:30Z <p>Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of contiunous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)\subset O$, where $K$ is any compact subset of $X$, and $O$ is any open subset of $Y$. So a basis of open sets is given by the following subsets: $[K_1,\dots,K_n,O_1,\dots,O_n]=[K_1,O_1 ]\cap\dots\cap [K_n,O_n]$, the collection of continuous maps $f:X\rightarrow Y$ that send each $K_i$ into $O_i$ for some specified collcetion of compact $K_i$'s and open $O_i$'s.</p> <p>This topology has some nice properties: the exponential law holds under some hypotheses on the spaces $X$ and $Y$, and is certainly true if all spaces involved are locally compact Hausdorff spaces, as will be the case from now on.</p> <p>My question is as follows: if $X$ is a locally compact Hausdorff space (or even a topological manifold), the compact open topology induces a topology on the set of homeomorphisms of $X$, which is a group. Does this topology turn $\mathrm{Homeo}(X)$ into a topological group? I can show that the product (composition) is continuous, but is the inverse too? $(f\rightarrow f^{-1})$</p> <p>I was able to prove continuity for compact spaces, where it is very easy to establish. I also managed to prove it for $X=\mathbb{R}$ because all homeomorphisms of $\mathbb{R}$ are monotone, but that's everything so far.</p> <p>I tried looking it up in several textbooks on topology and algebraic topology where the C.O. topology is usually discussed, but couldn't find a discussion on this topic anywhere.</p> http://mathoverflow.net/questions/58690/compact-open-topology-on-mathrmhomeox/58694#58694 Answer by Tom Ellis for Compact open topology on $\mathrm{Homeo}(X)$ Tom Ellis 2011-03-16T23:13:30Z 2011-03-16T23:13:30Z <p>The following article gives you a lot of information on the question you are asking:</p> <p>On Homeomorphism Groups and the Compact-Open Topology, Jan J. Dijkstra</p> <p><a href="http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf" rel="nofollow">http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf</a></p> <p><a href="http://www.jstor.org/pss/30037630" rel="nofollow">http://www.jstor.org/pss/30037630</a></p> <p>The answer is in general "no".</p> http://mathoverflow.net/questions/58690/compact-open-topology-on-mathrmhomeox/58725#58725 Answer by Robert Israel for Compact open topology on $\mathrm{Homeo}(X)$ Robert Israel 2011-03-17T06:11:18Z 2011-03-17T06:11:18Z <p>For a simple counterexample, let X be the subspace of R consisting of 0 and exp(n) for all integers n, and consider the homeomorphisms $f_n$ of X defined by $f_n(0) = 0$, $f_n(\exp(k)) = \exp(k-1)$ for $k \le -n$ or $k > n$, $\exp(k)$ for $-n &lt; k &lt; n$, and $\exp(-n)$ for $k = n$. Then $f_n$ converges to the identity map in the compact-open topology, but $f_n^{-1}$ does not: $f_n^{-1}$ is not in the neighbourhood of the identity map given by $K = X \cap [0, 1]$ and $U = X \cap (-1, 2)$ for any $n \ge 1$, because $\exp(-n) \in K$ and $f_n^{-1}(\exp(-n)) = \exp(n) \notin U$. </p> http://mathoverflow.net/questions/58690/compact-open-topology-on-mathrmhomeox/61400#61400 Answer by Abel Stolz for Compact open topology on $\mathrm{Homeo}(X)$ Abel Stolz 2011-04-12T13:37:57Z 2011-04-12T13:37:57Z <p>I don't know if this helps, but you could consider the set $[K,O]'$ of all maps $f\in Homeo(X)$ such that $f(K)\subset O$ and $f^{-1}(K)\subset O$. With the sets $[K,O]'$ as a subbase of neighborhoods of the identity, $Homeo(X)$ becomes a topological group, if $X$ is a locally compact $T_3$-space. This topology is called the Braconnier topology. It coincides with the compact open topology whenever $X$ is compact, or locally connected. In general it is stronger than the compact open topology.</p> <p>Unfortunately I don't know any references for this facts, except the lecture notes of the 1982/83 lectures of Prof. Holdgruen on harmonic analysis (in a shelf in the library of the mathematics institute in Goettingen), because I learned this in my undergrad courses and not from books.</p> http://mathoverflow.net/questions/58690/compact-open-topology-on-mathrmhomeox/61430#61430 Answer by Johannes Ebert for Compact open topology on $\mathrm{Homeo}(X)$ Johannes Ebert 2011-04-12T17:34:30Z 2011-04-12T17:34:30Z <p>R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946) 593–610.</p> <p>If $X$ is locally compact and \emph{locally connected} (!), then $Homeo(X)$ is a topological group.</p>