Regular spaces that are not completely regular - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:13:38Z http://mathoverflow.net/feeds/question/17371 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17371/regular-spaces-that-are-not-completely-regular Regular spaces that are not completely regular Michał Kukieła 2010-03-07T10:38:50Z 2013-04-17T18:01:13Z <p>In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. Later I was told by a colleague that such examples are rare and difficult to construct.</p> <p>I know there is an example of such a space, called the Tychonoff corkscrew (or the spiral staircase), in the "<em>Counterexamples in topology</em>" book by Steen and Seebach. I've also found the following paper, though at the moment I'm not able to view it: A.B. Raha "<em>An example of a regular space that is not completely regular</em>", Proceedings Mathematical Sciences 102 (1992), 49-51.</p> <p>Are there any other, folklore examples of regular spaces that are not completely regular? Are there any relatively easy ones?</p> http://mathoverflow.net/questions/17371/regular-spaces-that-are-not-completely-regular/17380#17380 Answer by Georges Elencwajg for Regular spaces that are not completely regular Georges Elencwajg 2010-03-07T14:17:44Z 2010-03-07T14:17:44Z <p>Dear Michal, Munkres presents a regular space that is not completely regular as a very detailed exercise (more than half a page!) to §33 in his book "Topology, Second Edition, Prentice Hall,2000" (page214, exercise 11).</p> <p>It is a different example from that in Steen and Seebach (or Dugundji for that matter), in that it doesn't use ordinal numbers. I don't know the Polish educational system, but this might be an advantage for undergraduates not yet knowing these ordinals. Also I'd like to advertise Munkres's book which is a real gem (though I'm sure you have excellent topology books, given the brilliant Polish tradition in that field).</p> http://mathoverflow.net/questions/17371/regular-spaces-that-are-not-completely-regular/17382#17382 Answer by Clark Barwick for Regular spaces that are not completely regular Clark Barwick 2010-03-07T14:44:58Z 2010-03-07T14:44:58Z <p>These examples seem to be very difficult to construct. The problem is that any local compactness or uniformity will automatically boost your space to a Tychonoff space, and Tychonoff spaces are closed under passing to subspaces or products. Consequently, there's doesn't seem to be a "machine" for producing these kinds of spaces.</p> <p>The idea of all the counterexamples $X$ is to write down enough open sets of $X$ to make it clear that points can be separated from closed subsets, but to somehow rig things so that any continuous real-valued function on $X$ identifies two distinct points of the space.</p> <p>The example in Munkres's textbook that Elencwajg mentions is a pretty straightforward one (relatively speaking); it's the same in spirit as Raha's example, which is the easiest I've found. Here it is:</p> <p>For every even integer $n$, set $T_n:={n}\times(-1,1)$, and let $X_1=\bigcup_{n\textrm{ even}}T_n$. Now let $(t_k)_{k\geq 1}$ be an increasing sequence of positive real numbers converging to $1$.</p> <p>For every odd integer $n$, set <code>$$T_n:=\bigcup_{k\geq 1}\{(x,y)\in\mathbf{R}^2\ |\ (x-n)^2+y^2=t_k^2\}$$</code> and let $X_2=\bigcup_{n\textrm{ odd}}T_n$. Now let <code>$$X=\{a,b\}\cup\bigcup_{n\in\mathbf{Z}}T_n$$</code></p> <p>Topologize $X$ so that:</p> <ol> <li>every point of $X_2$ except the points $(n,t_k)$ are isolated;</li> <li>a neighborhood of $(n,t_k)$ consists of all but finitely many elements of <code>$\{(x,y)\in\mathbf{R}^2\ |\ (x-n)^2+y^2=t_k^2\}$</code>;</li> <li>a neighborhood of a point $(n,y)\in X_1$ consists of all but a finite number of points of <code>$\{(z,y)\ |\ n-1&lt;z&lt;n+1\}\cap(T_{n-1}\cup T_n)$</code>;</li> <li>a neighborhood of $a$ is a set $U_c$ containing $a$ and all points of $X_1\cup X_2$ with $x$-coordinate greater than a number $c$;</li> <li>a neighborhood of $b$ is a set $V_d$ containing $b$ and all points of $X_1\cup X_2$ with $x$-coordinate less than a number $d$.</li> </ol> <p>This is a space that is $T_3$, but every continuous map $f:X\to\mathbf{R}$ has the property that $f(a)=f(b)$, so it is not $T_{3\frac{1}{2}}$.</p> http://mathoverflow.net/questions/17371/regular-spaces-that-are-not-completely-regular/49673#49673 Answer by LostInMath for Regular spaces that are not completely regular LostInMath 2010-12-16T18:19:14Z 2010-12-16T18:19:14Z <p>Raha's article is available (for free) at the home page of The Proceedings of the Indian Academy of Sciences – Mathematical Sciences (link: <a href="http://www.ias.ac.in/mathsci/index.html" rel="nofollow">http://www.ias.ac.in/mathsci/index.html</a>).</p> http://mathoverflow.net/questions/17371/regular-spaces-that-are-not-completely-regular/127873#127873 Answer by Ramiro de la Vega for Regular spaces that are not completely regular Ramiro de la Vega 2013-04-17T18:01:13Z 2013-04-17T18:01:13Z <p>I believe the simplest example is given in:</p> <blockquote> <p>A. Mysior, <em>"A regular space that is not completely regular"</em>, PAMS 81 (1981), No.4, 652-653. </p> </blockquote>