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 $L_0$.
Every basis neighborhood of $X$ is closed as well as open, so $X$ is zero dimensional and therefore regular since it is clearly T1.
$X$ is also completely regular since if $C$ is a closed set and $p \notin C$, ...