Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named *Gutik's hedgehog*. It is homeomorphic to the space
$$H:=\{(0,0)\}\cup\{(\tfrac1n,0):n\in\mathbb N\}\cup\{(\tfrac1n,\tfrac1{nm}):n,m\in\mathbb N\},$$ endowed with the topology $\tau$ consisting of sets $U\subset H$ satisfying the following two conditions:

(1) if $(\frac1n,0)\in U$ for some $n\in\mathbb N$, then there exists $m\in\mathbb N$ such that $(\frac1n,\frac1{nk})\in U$ for all $k\ge m$;

(2) if $(0,0)\in U$, then there exists $m\in\mathbb N$ such that $(\frac1n,\frac1{nk})\in U$ for all $n\ge m$ and all $k\in\mathbb N$.

It turns out that Gutik's hedgehog is a test space for regularity in the class of first-countable Hausdorff spaces.

**Theorem.** *A first-countable Hausdorff space is regular if and only if it contains no topological copies of the Gutik hedgehog.*

Because of this fundamental role in testing regularity, I admit that Gutik's hedgehog is known in topology under some different name. I would be grateful for any information in this respect.

**Remark 1.** The Gutik's hedgehog resembles (but is not equal to) the non-regular space of Smirnov, see Example 64 in "Counterexamples in Topology".