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?



Regular spaces that are not completely regular
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 $T\_{3\frac{1}{2}}$.


