While studying some examples of regular spaces which are not completely regular, I came across Steen's and Seebach's "Counterexamples in Topology". In this book, after searching for examples, I only found four examples of regular not completely regular spaces: Tychonoff Corkscrew, deleted Tychonoff Corkscrew, Hewitt's Condensed Corkscrew and Thomas' Corkscrew.
All this examples are variations of the same example: the Tychonoff Corkscrew. So my question is the following one: is there a regular not completely regular space which is not a corkscrew? Or in other way, is it possible to define the property of being a corkscrew is some sense in order to have a result of the form that every regular space is completely regular except if it is a corkscrew?
Alternatively, is it possible to construct a regular not completely regular space which is not a corkscrew in the sense of being radically different -in the sense that use totally different ideas and it is not homeomorphic to the cited ones- from the previous examples?
Best Answer
The following example (which I am fairly certain is not a corkscrew-like construction) is essentially taken from
Let $X = (\;\mathbb{R} \times [ 0 , 2 )\;) \cup \{ \langle 0 , -1 \rangle \}$, and topologise $X$ as follows:
It is fairly easy to check that $X$ is regular, the basic idea being that the basic open neighbourhoods of the points in $\mathbb{R} \times [ 0 , 2 )$ described above are actually clopen, and $\overline{U_{n+2}} \subseteq U_n$ for all $n \in \mathbb{N}$.
However there is no continuous $f : X \to [0,1]$ such that $f(0,-1) = 0$ and $F = \{ \langle x , 0 \rangle \in X : 0 \leq x \leq 1 \} \subseteq f^{-1} [ \{ 1 \} ]$. The basic idea here is to show that if $F \subseteq f^{-1} [ \{ 1 \} ]$, then $f( 0 , -1 ) = 1$. For this we first prove by induction that $B_n = \{ x \in ( n-1 , n ] : f( x , 0 ) = 1 \}$ is infinite for all $n \geq 1$.
Now note that if $f ( 0,-1 ) \neq 1$, then there would be an $n \geq 1$ such that $U_n \subseteq f^{-1} [\;[0,1)\;]$, however as $B_{n+1} \subseteq U_n$ this is impossible.