Can a connected planar compactum minus a point be totally disconnected? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:44:16Z http://mathoverflow.net/feeds/question/16578 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16578/can-a-connected-planar-compactum-minus-a-point-be-totally-disconnected Can a connected planar compactum minus a point be totally disconnected? HW 2010-02-27T02:38:23Z 2010-02-28T00:28:47Z <p>What the title said. In a slightly more leisurely fashion:-</p> <blockquote> <p>Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let <code>$x\in X$</code>. Can $X\smallsetminus{x}$ be totally disconnected?</p> </blockquote> <p>Note that the <a href="http://en.wikipedia.org/wiki/Knaster%E2%80%93Kuratowski_fan" rel="nofollow">Knaster-Kuratowski fan</a> shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.</p> <p>To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.</p> http://mathoverflow.net/questions/16578/can-a-connected-planar-compactum-minus-a-point-be-totally-disconnected/16627#16627 Answer by Anton Petrunin for Can a connected planar compactum minus a point be totally disconnected? Anton Petrunin 2010-02-27T19:42:35Z 2010-02-27T19:56:19Z <p>Let denote by $U_n\subset \mathbb R^2$ a sequence of open bounded neigborhoods of $X$, so that $$U_{n+1}\subset U_n\ \ \text{and}\ \ \bigcap_n U_n=X.$$ We can assume that all $U_n$ are connceted and therefore path-connected. Coose a point $p\in X$ distict from $x$ and consider a sequence of paths $\gamma_n$ in $U_n$ from $p$ to $x$. Fix $\epsilon>0$ such that $\epsilon&lt;|p-x|$. For each path choose the smalest value $t_n\in[0,1]$ so that $|\gamma_n(t_n)-x|=\epsilon$. The image $Z_n=\gamma([0,t_n])$ is connected compact set. Let $Z$ be a Hausdorff limit of a subsequence of $Z_n$. Note that $Z$ is a compact connected subset of $X$. Clearly, $Z\not\ni x$ and it contains at least two points; a contradiction</p> http://mathoverflow.net/questions/16578/can-a-connected-planar-compactum-minus-a-point-be-totally-disconnected/16630#16630 Answer by Bill Johnson for Can a connected planar compactum minus a point be totally disconnected? Bill Johnson 2010-02-27T20:04:08Z 2010-02-27T20:04:08Z <p>Being planar has nothing to do with the problem. Suppose a totally disconnects $X$ and choose $b$ different from $a$. By passing to a sub continuum, assume that no proper sub continuum contains both $a$ and $b$. Take non empty disjoint open sets $U$ and $V$ whose union is $X\sim a$. WLOG $b$ is in $U$, and observe that $U\cup {a}$ is closed and connected. </p> http://mathoverflow.net/questions/16578/can-a-connected-planar-compactum-minus-a-point-be-totally-disconnected/16645#16645 Answer by Jonas Meyer for Can a connected planar compactum minus a point be totally disconnected? Jonas Meyer 2010-02-28T00:28:47Z 2010-02-28T00:28:47Z <p>Two great answers have already been given, and I don't claim to add much, but here is something anyway.</p> <p>A totally disconnected locally compact Hausdorff space has a basis of clopen sets, according to <a href="http://books.google.com/books?id=B785AETmFKEC&amp;lpg=PR1&amp;pg=PA136#v=onepage&amp;q=&amp;f=true" rel="nofollow">Proposition 3.1.7 of Arhangel'skii and Tkachenko</a>, for example. A closed set in <code>$X-\{a\}$</code> need not be closed in $X$, but if $X$ is a metric space then the clopen subsets of <code>$X-\{a\}$</code> at positive distance to $a$ will be clopen in $X$. Thus if $X$ is a compact metric space with more than one point and <code>$X-\{a\}$</code> is totally disconnected, then $X$ is not connected.</p>