Are all compactly generated Hausdorff spaces regular?

Question

If X is a compactly generated Hausdorff space then I want to know if X is a regular space? (we can not expect X to be normal because there are examples of locally compact Hausdorff space that fails to be normal and any locally compact space is compactly generated) .

Answer 1

Not necessarily. For instance, let $X$ be a closed disk, let $D$ be its interior, and topologize $X$ as follows. Say a subset $A\subseteq X$ is open iff $A\cap (D\cup\{x\})$ is open in $D\cup\{x\}$ with respect to the usual topology for all $x\in X\setminus D$. This topology is finer than the usual topology on the disk, so it is Hausdorff. It is not regular, since if $x\in X\setminus D$ then $X\setminus (D\cup \{x\})$ is closed but $x$ and $X\setminus(D\cup\{x\})$ cannot be separated with open sets. Finally, $X$ is compactly generated, since it is the colimit of the spaces $D\cup\{x\}$ with their usual topologies, which are each compactly generated. (In fact, $X$ is first-countable: if $\{U_n\}$ is a local base at a point $x$ in the usual topology, then $\{U_n\cap(D\cup\{x\})\}$ is a local base at $x$ in the topology of $X$.)