# Is Thomas' Corkscrew completely regular? (from Counterexamples in Topology)

In Example 94 of Counterexamples in Topology. In the example itself, it is written that the space is completely regular. But in the appendix at the end of the book, it is written that the space is regular but not completely regular. Which one is correct?

Thank you!

94. Thomas' Corkscrew

Let $X = \bigcup_{i=0}^\infty L_i$ be the union of lines in the plane where $L_0 = \{ (x,0) \mid x \in (0,1) \}$, and for $i \geq 1$, $L_i = \{ ( x , \frac{1}{i} \mid x \in [0,1) \}$. If $i > 0$, each point of $L_i$ except for $(0,\frac{1}{i})$ is open; basis neighborhoods of $(0,\frac{1}{i})$ are subsets of $L_i$ with finite complements. Similarly, the sets $U_i(x,0) = \{ (x,0) \} \cup \{ (x,\frac{1}{n}) \mid n > i \}$ form a basis for the points in $L_0$.

1. Every basis neighborhood of $X$ is closed as well as open, so $X$ is zero dimensional and therefore regular since it is clearly T1.

2. $X$ is also completely regular since if $C$ is a closed set and $p \notin C$, ...

5 If we use copies of Thomas' plank to build a corkscrew (as in the Tychonoff corkscrew) we can obtain a regular space which is not Urysohn, since every continuous function $f$ will be constant except for a countable set on coordinate axes at each level of the corkscrew. Thus if $p^+$ and $p^-$ are the infinity points, $f(p^+) = f(p^-)$.
Note that this implies that Thomas' Corkscrew is not completely regular, since $\{ p^- \}$ would be a closed set not containing $p^+$ which cannot be separated from $p^+$ by a continuous real-valued function.