A regular space which is not completely regular: Mysior’s example.

general-topology

In this paper https://www.ams.org/journals/proc/1981-081-04/S0002-9939-1981-0601748-4/S0002-9939-1981-0601748-4.pdf, Mysior gives a example of a regular space which is not completely regular, but I didn't understand the construction of space (X). Is there any easier way to visualize, such as a picture?

The description is quite simple (it's two paragraphs); the set of points is just $$\{(x,y)\in \Bbb R^2\mid y \ge 0\} \cup \{a\}$$, where $$a$$ is some point outside the plane (sort of infinity, one could say).

All points in the plane part with $$y>0$$ are isolated (so lots of scattered dust there) and the "glue" is provided by points of the form $$(x,0)$$. A basic neighbourhood of $$(x,0)$$ is of the form

$$U((x,0),F) = \{(x,0)\} \cup \left( I_x:=\{(x,y)\mid 0 \le y < 2\} \cup I'_x:=\{(x,x+y)\mid 0 \le y < 2\right) \setminus F$$

where $$F \subseteq I_x\cup I'_x$$ is finite. $$I_x$$ is a fixed length vertical open-ended line segment emanating from $$(x,0)$$ and $$I'_x$$ is an angle-45-degrees open-ended line segment emanating from $$(x,0)$$ to the right side. The substracted finite set is just to ensure that the space will be Hausdorff as otherwise we could not separate $$(x,0)$$ from $$(x,1)$$ by open sets and now we can, by using $$F=\{(x,1)\}\subseteq I_x$$ e.g.

A basic neighbourhood of $$a$$ looks like $$U_n(a)= \{a\} \cup \{(x,y)\mid y \ge 0, x > n\}$$ where $$n$$ varies over $$\Bbb N$$ (so $$a$$ lies "to the right" of the half plane.

All this is nice and consistent (e.g. all $$U_n(a)$$ consist of interior points only when we consider the points in it, and likewise for the sets $$U((x,0),F)$$) so in total this defines all open sets in the usual way.

So visually it's not that hard: scattered upper plane points that can be in two-pronged open sets for the axis points and one infinity point on the right.

Hope this helps a little bit.