Questions like these are easily answered with a search of pi-Base:

π-Base, Search for $k_3$-space+Paracompact+~$T_2$

Six counterexamples are listed there today, including Tyrone's example. I'll suggest $\omega_1+1$ with the endpoint doubled (S37) as another. It is compact and therefore paracompact. It is compactly generated because each point has a compact Hausdorff neighborhood.

In fact, this is a general construction: take your favorite paracompact Hausdorff locally compact space, then double a non-isolated point.

Questions like these are easily answered with a search of pi-Base:

π-Base, Search for $k_3$-space+Paracompact+~$T_2$

Six counterexamples are listed there today, including Tyrone's example. I'll suggest $\omega_1+1$ with the endpoint doubled (S37) as another. It is compact and therefore paracompact. It is compactly generated because each point has a compact Hausdorff neighborhood.

In fact, this is a general construction: take your favorite paracompact Hausdorff locally compact space, then double a non-isolated point.

Questions like these are easily answered with a search of pi-Base:

π-Base, Search for $k_3$-space+Paracompact+~$T_2$

Six counterexamples are listed there today, including Tyrone's example. I'll suggest $\omega_1+1$ with the endpoint doubled (S37) as another. It is compact and therefore paracompact. It is compactly generated because each point has a compact Hausdorff neighborhood.

Questions like these are easily answered with a search of pi-Base:

π-Base, Search for $k_3$-space+Paracompact+~$T_2$

Six counterexamples are listed there today, including Tyrone's example. I'll suggest $\omega_1+1$ with the endpoint doubled (S37) as another. It is compact and therefore paracompact. It is compactly generated because each point has a compact Hausdorff neighborhood.

In fact, this is a general construction: take your favorite paracompact Hausdorff locally compact space, then double a non-isolated point.