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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.

I know there is an example of such a space, called the Tychonoff corkscrew (or the spiral staircase), in the "Counterexamples in topology" 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 "An example of a regular space that is not completely regular", Proceedings Mathematical Sciences 102 (1992), 49-51.

Are there any other, folklore examples of regular spaces that are not completely regular? Are there any relatively easy ones?

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If you use the chart in the back of the book, you can find all of Steen & Seebach's examples that meet your criteria. –  Gerald Edgar Mar 7 '10 at 14:07
    
I haven't read the examples in Steen & Seebach very thoroughly. However, they all seem to be based on the Tychonoff corkscrew construction. –  Michał Kukieła Mar 7 '10 at 14:49
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4 Answers

up vote 10 down vote accepted

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.

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.

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:

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 $1$.

For every odd integer $n$, set $$T_n:=\bigcup_{k\geq 1}\{(x,y)\in\mathbf{R}^2\ |\ (x-n)^2+y^2=t_k^2\}$$ and let $X_2=\bigcup_{n\textrm{ odd}}T_n$. Now let $$X=\{a,b\}\cup\bigcup_{n\in\mathbf{Z}}T_n$$

Topologize $X$ so that:

  1. every point of $X_2$ except the points $(n,t_k)$ are isolated;
  2. a neighborhood of $(n,t_k)$ consists of all but finitely many elements of $\{(x,y)\in\mathbf{R}^2\ |\ (x-n)^2+y^2=t_k^2\}$;
  3. a neighborhood of a point $(n,y)\in X_1$ consists of all but a finite number of points of $\{(z,y)\ |\ n-1<z<n+1\}\cap(T_{n-1}\cup T_n)$;
  4. 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 $c$;
  5. 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 $d$.

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 $T_{3\frac{1}{2}}$.

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(+1) Not all counterexamples will be such that continuous real-valued functions identify two distinct points. IIRC, the Deleted Tychonoff Corkscrew is regular and completely Hausdorff, but not completely regular. –  Cameron Buie Sep 16 '13 at 1:50
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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).

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).

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Raha's article is available (for free) at the home page of The Proceedings of the Indian Academy of Sciences – Mathematical Sciences (link: http://www.ias.ac.in/mathsci/index.html).

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Looks like I can only download the first page. –  Michał Kukieła Jan 5 '11 at 22:11
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All 3 pages are downloadable, but separately: ias.ac.in/jarch/mathsci/102/00000054.pdf ias.ac.in/jarch/mathsci/102/00000055.pdf ias.ac.in/jarch/mathsci/102/00000056.pdf \\However, the copy available from springer dx.doi.org/10.1007/BF02837178 is better - it is OCRed. –  Martin Sleziak Oct 11 '11 at 11:13
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I believe the simplest example is given in:

A. Mysior, "A regular space that is not completely regular", PAMS 81 (1981), No.4, 652-653.

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