gn.general topology - 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 $X_1=bigcup_{ntextrm{ even}}T_n$. Now let $(t_k)_{kgeq 1}$ be an increasing sequence of positive real numbers converging to $1$.

For every odd integer $n$, set $$T_n:=bigcup_{kgeq 1}{(x,y)inmathbf{R}^2 | (x-n)^2+y^2=t_k^2}$$ and let $X_2=bigcup_{ntextrm{ odd}}T_n$. Now let $$X={a,b}cupbigcup_{ninmathbf{Z}}T_n$$

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 ${(x,y)inmathbf{R}^2 | (x-n)^2+y^2=t_k^2}$;


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 $X_1cup X_2$ with $x$-coordinate greater than a number $c$;


5. a neighborhood of $b$ is a set $V_d$ containing $b$ and all points of $X_1cup X_2$ with $x$-coordinate less than a number $d$.

This is a space that is $T_3$, but every continuous map $f:Xtomathbf{R}$ has the property that $f(a)=f(b)$, so it is not $T_{3frac{1}{2}}$.